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Sequences And Series - Class 11 Engineering Maths - Extra Questions

 The sum of the infinite series 1+1+22!+1+2+223!+1+2+22+234!+... is eyex Find x+y2



Write four more numbers in the following pattern-
 132639412..........



Find the first 3 terms of a G.P. if a=4 and r=2.



Find the sum of all natural numbers from 1 to 200 which are divisible by 4.



Find the arithmetic mean of 4 and 6.



Find the common ratio and the general term of the following geometric sequences.
0.02, 0.006, 0.0018, .....



A.M. of a-2, a, a+2 is ____.



A water tank has steps inside it. A monkey is sitting on the top most step. (ie, the first step) The water level is at the ninth step.
(a) He jumps 3 steps down and then jumps back 2 step up. In how many jumps will he reach the water level?
(b) After drinking water, he wants to go back. For this, he jumps 4 steps up and then jumps back 2 steps down in every move. In how many jumps will he reach back the top step?
616491_f59be177db5e474fa7fe4cc05fc57697.png



A.M. of 1, 2, x, 3 is 0, then x= _____.



Complete the following patterns:
(i) 40, 35, 30, __, __, __
(ii) 0, 2, 4, ___ , ___ , ___ .
(iii) 84, 77, 70, ___ , ___ , ___ .
(iv) 4.4, 5.5, 6.6, ___ , ___ , ___ .
(v) 1, 3, 6, 10, ___ , ___ , ___ .
(vi) 1, 1, 2, 3, 5, 8, 13, 21, ___ , ___ , ___ . (This sequence is called FIBONACCI SEQUENCE)
(vii) 1, 8, 27, 64, ___ , ___ , ___ .



A geometric series consists of four terms and has a positive common ratio. The sum of the first two terms is 8 and the sum of the last two terms is 72. Find the series



Find the sum of n terms of the series \left(4-\displaystyle\frac{1}{n}\right)+\displaystyle \left(4-\frac{2}{n}\right)+\displaystyle \left(4-\frac{3}{n}\right)+........



Write the smallest 7-digits largest and smallest number having three different digits.



\dfrac{1}{{1.2.3}} + \dfrac{1}{{2.3.4}} + \dfrac{1}{{3.4.5}} + \,..... + \dfrac{1}{{n\left( {n + 1} \right)\left( {n + 2} \right)}} = \dfrac{{n\left( {n + 3} \right)}}{{4\left( {n + 1} \right)\left( {n + 2} \right)}}



Solve: \displaystyle \sum_{n = 1}^{13}(t^n - t^{n + 1}) =



If \alpha ,\,\beta \,,\gamma are in A.P. show that \cot\,\beta  = \dfrac{{\sin \alpha  - \sin \gamma }}{{\cos \gamma  - cos\alpha }}.



complete the table . what do you notice about the numbers in the orange band ?
1713
2814
3915
41016
51117
612



{ a }_{ 1 }=5;{ a }_{ 4 }=9\dfrac { 1 }{ 2 } find { a }_{ 2 },{ a }_{ 3 }



If,
            1, 5, 8=76
            2, 7, 3=25
            3, 4, 9=89
            4, 5, 7=69
THEN 
              5, 3, 8=?



If the mean of 20,\,14,\,16,\,19,\,p and 21 is 27 then find the value of 'p'



The first term of a GP isThe sum of the third and fifth term isFind the common ratio of GP.



13,25,14,23,15,21,16,?,?
1) 14,11
2) 19,17



Look at the following matchstick pattern of letter A, the A's are not separate. Two neighbouring A's have two common matchsticks. Observe the pattern and find the rule that gives the number of matchsticks.
1124650_e9dcb77c80454c39b4c60e318731676e.jpg



If a,b,c are in AP and b,c,d are in GP and \dfrac{1}{c},\dfrac{1}{d}, \dfrac{1}{e} are in AP, prove that a,c,e are in GP.



Find the sum of series 1+2+3+4+5+6+...+n.



Calculate the sum of the series 1+3+5+7+....2n-1.



Find the {15^{th}} term of the series 3,9,15,21,27,33,....



Find the sum to n terms of the series
5+11+19+29+......



Find the sum of terms of the sequence till n terms : 6+9+16+27+42+....+n terms.



Find the average of the first five multiples of 10.



Find the missing number in the series : 
40 , 54 , 82 , ? , 180 , 250



Write opposites of the following :
30\ km north



If 18, a, b, -3 are in AP then find a+b.



Solve: 2-\dfrac{3}{5}.



Find the term of the series 25, 22\dfrac{3}{4}, 20\dfrac{1}{2},18\dfrac{1}{4}..... which is numerically smallest.



Solve for a. 
\frac{3a-2}{7}-\frac{a-2}{4}=2, 



If { 1 }^{ 2 }+{ 2 }^{ 3 }+{ 3 }^{ 2 }+.....+\left( 2003 \right) ^{ 2 }=\left( 2003 \right) \left( 4007 \right) \left( 334 \right) and\left( 1 \right) \left( 2003 \right) +\left( 2 \right) \left( 2002 \right) +\left( 3 \right) \left( 2001 \right) =\left( 2003 \right) \left( 334 \right) \left( x \right) , then x equals



The mean of first 10 whole numbers is



Find three consecutive odd numbers whose sum is 147.



Find three consecutive even number whose sum is 234.



5,10,15,20,...



Prove that \sum\limits_{n=1}^{\infty}\dfrac{1}{n!}=e-1.



Fill in the blanks:
\dfrac {2}{8} \times 3 = \dfrac {2}{8} + -+-+-



Ia, b, c are is G.P and a - b, c - a and b - a are in H.P, then a + 4b + c is equal to 



Write the solution set of \left|x+\dfrac{1}{x}\right|>2.



Find three consecutive odd numbers whose sum is 234.



1,2,4,8,16,....



If a,b,c,d are in GP, show that
(b-c)^2+(c-a)^2+(d-b)^2=(a-d)^2



Fill in the blanks in the expressions with the proper numbers.
(5\ \times 4) > (7\times \Box )



If a,b,c are in GP and a^{1/x}=b^{1/y}=c^{1/z}, prove that x,y,z are in AP.



The sum of the first 20 terms of the series 1+\dfrac{3}{2}+\dfrac{7}{4}+\dfrac{15}{8}+\dfrac{31}{16}+...., is



x = 1 + a + a ^ { 2 } + \ldots \infty , y = 1 + b + b ^ { 2 } + \ldots \infty   then   1 + a b + a ^ { 2 } b ^ { 2 } + \ldots \infty =



P and Q are brothers, X and Y are sisters, son of P is the brother of Y. How is Q related to X ?



The Fibonacci sequence is defined by 1=a_{1}=a_{2} and a_{n}=a_{n-1}+a_{n-1},n > 2.
Find \dfrac {a_{n}}{a_{n}+1} for n=1,2,3,4.



Identify whether the following sequence is a geometric sequence or not.
-3,9,-27,81



Identify whether the following sequence is a geometric sequence or not.
2,6,18,54



If the heights of 5 persons are 144 cm, 152 cm, 151 cm, 158 cm, and 155 cm respectively. Find the mean height.



Fill the squares with all the numbers from 46 to 54

rule:- total of each line should be 150.
49
46
5247



The mean of 45,84,75,87,47 is  



Identify whether the following sequence is a geometric sequence or not.
1,4,9,16



Identify whether the following sequence is a geometric sequence or not.
\dfrac{1}{2},\dfrac{2}{4},\dfrac{4}{8},\dfrac{8}{16}



Identify whether the following sequence is a geometric sequence or not.
\dfrac{1}{2},\dfrac{2}{3},\dfrac{3}{4},\dfrac{4}{5}



Find the value of question mark .
1392131_7b990600609e4bf0ac1a8f791925356d.png



Identify whether the following sequence is a geometric sequence or not.
25,5,1,\dfrac{1}{5}



The fourth term of a G.P. is 8 and the 7th term is 64, find the G.P.



Find the mean of 43, 51, 50, 57 and 54.



Find the sum of the following series:
0.6+0.66+0.666+... to n terms.



Increased by 3



The sum of the following series to infinity \displaystyle \frac{1}{1.3.5} +\frac{1}{3.5.7} +\frac{1}{5.7.9} +\,...\,\infty  is   \dfrac{1}{3c} .Find c 



 The sum of the series \displaystyle \frac2{3!}+\frac4{5!}+\frac6 {7!}+... to \infty = \displaystyle \frac{a}{e}. Find (a+3)^2.



If A_{n}= (n, n + 1) then 10(A_{10}A_{11})^{2}+11(A_{11}+A_{12})^{2}+......+20(A_{20}A_{21})^{2} is equal to



Sum of the series \displaystyle 1+\left ( 1+2 \right )+\left ( 1+2+3 \right )+\left ( 1+2+3+4 \right )+... to n terms be \displaystyle \frac{1}{m}n\left ( n+1 \right )\left [ \frac{2n+1}{k}+1 \right ] .Find the value  of  k+m.



Considre the sequence \displaystyle \left ( a_{n} \right )n\geq 0 given by the following relation:\displaystyle a_{0}=4,a_{1}=22, and for all \displaystyle n\geq 2, \displaystyle a_{n}=6a_{n-1}-a_{n-2}. Prove that there exist sequences of positive integers \displaystyle \left ( x_{n} \right )n\geq 0,\left ( y_{n} \right )n\geq 0 such that\displaystyle a_{n}=\frac{y^{2}_{n}+7}{x_{n}-y_{n}}, for all \displaystyle n\geq 0.



If the set natural numbers, is partitioned into subsets \displaystyle S_{1}= \left \{ 1 \right \},S_{2}= \left \{ 4,5,6 \right \},S_{4}= \left \{ 7,8,9,10 \right \} The last terms of these groups is 1,1+2,1+2+3,1+2+3+4 ...... Find the sum of the elements in the subset S_{50}.



sum the series 1+2+...…….+100



sum of the series is 1+3+6+10+15+... to n terms  \displaystyle \frac{n\left ( n+m \right )\left ( n+p \right )}{k}.Find k-m-p ?



Sum of the series 4+6+9+13+18+... to n terms be \displaystyle =\frac{n}{k}\left ( n^{2}+3n+m \right ).Find  m-k ?



Find the sum of 2n terms of the series \displaystyle 5^{3}+4.6^{3}+7^{3}+7^{3}+4.8^{3}+9^{3}+4.10^{3}+....



If \displaystyle s_{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...\frac{1}{n},\left ( n\: \epsilon \: N \right ), then \displaystyle s_{1}+s_{2}+s_{3}+s_{4}+...s_{n}=\left ( n+\lambda  \right )s_{n+1}-\left ( n+1 \right ). Find the value of \displaystyle \lambda



Find the sum to n terms of the sequence 8, 88, 888, 8888, .... .



If a,b,c and d are in G.P. show that \displaystyle ({ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 })({ b }^{ 2 }+{ c }^{ 2 }+{ d }^{ 2 })={ (ab+bc+cd) }^{ 2 }.



If {S}_{n}={(4)}^{n}+3, what is the units digit of {S}_{100}?



Rahul's father wants to deposit some amount of money every year on the day of Rahul's birthday. On his 1st birth day Rs.100, on his 2nd birth day Rs.300, on his 3rd birth day Rs.600, on his 4th birthday Rs.1000 and so on. 
What is the amount deposited by his father on Rahul's 15th birthday.



The product of two numbers is 119 and their AM is 12. Find the numbers.



Find x, if the given numbers are in A.P. 5, (x - 1), 0



Find the last digit of 1^5 + 2^5 + ..... + 99^5.



The following geometric arrays suggest a sequence of numbers.
(a) Find the next two terms.
(b) Find the 100^{th} term.
(c) Find the n^{th} term.

569390_7cb46a82bdc441f7bec076e10a84f30b.png



Find the sum of all the digits of the result of the subtraction 10^{99} - 99.



The sum of the first n terms of a sequence is \dfrac {7^{n} - 6^{n}}{6^{n}}, Find its n^{th} term. Determine whether the sequence is A.P. or G.P



Find the sum to infinite terms of the series
\dfrac {7}{5} \left (1 + \dfrac {1}{10^{2}} + \dfrac {1.3}{1.2} \cdot \dfrac {1}{10^{4}} + \dfrac {1.3.5}{1.2.3} \cdot \dfrac {1}{10^{6}} + ......\right )



If the sum of the first n natural numbers is {S}_{1} and that of their squares is {S}_{2} and cubes is {S}_{3}, then show that 9{ S }_{ 2 }^{ 2 }={ S }_{ 3 }(1+8{ S }_{ 1 }).



The sum of first n terms of a sequence is \cfrac { { 6 }^{ n }-{ 5 }^{ n } }{ { 5 }^{ n } }
Find its n^{th} term and examine whether the sequence is an A.P of G.P.



Find the sum of the series 1\cdot 2+2\cdot 3+3\cdot 4+\cdots +n\left( n+1 \right) .



Find the { n }^{ th } convergent to \dfrac { 6 }{ 5- } \dfrac { 6 }{ 5- } \dfrac { 6 }{ 5- } \cdots .



If n is a multiple of 6, show that each of the series \displaystyle n-\frac{n(n-1)(n-2)}{\left\lfloor 3\right.}\cdot 3+\frac{n(n-1)(n-2)(n-3)(n-4)}{\left\lfloor 5\right.}\cdot 3^2-....., \displaystyle n-\frac{n(n-1)(n-2)}{\left\lfloor 3\right.}\cdot \frac{1}{3}+\frac{n(n-1)(n-2)(n-3)(n-4)}{\left\lfloor 5\right.}\cdot\frac{1}{3^2}-....., is equal to zero.



The 4^{th} term of a geometric sequence is \dfrac {2}{3} and the seventh term is \dfrac {16}{81}. Find the geometric sequence



Express \sqrt{19} as a continued fraction, and find a series of fractions approximating to its value.



Show that 1+\dfrac { 1 }{ 3\cdot { 2 }^{ 2 } } +\dfrac { 1 }{{ 2 }^{ 4 } } +\dfrac { 1 }{ 7.{ 2 }^{ 6 } } +....=\log _{ e }{ 3 } .



If { u }_{ 1 }=\dfrac { a }{ b } ,{ u }_{ 2 }=\dfrac { b }{ a+b } ,{ u }_{ 3 }=\dfrac { a+b }{ a+2b } ,\cdots , each successive fraction being formed by taking the denominator and the sum of the numerator and denominator of the preceding fraction for its numerator and denominator respectively, show that { u }_{ \infty  }=\dfrac { \sqrt { 5 } -1 }{ 2 } .



If a=\underset{55}{\underbrace{111.........1}}, b=1+10+10^2+10^3+10^4 and c=1+10^5+10^{10}+....+10^{50}, then show following.
(i) 'a' is a not composite number
(ii) a\ is\ not\ equal\ bc



If \sum_{r=0}^{n} \dfrac { (-1)^r \cdot C_r}{(r+1)(r+2)(r+3) } = \dfrac {1}{a(n+b) } , then a+b is :



Study the pattern:
91 \times 11 \times 1 = 1001
91 \times 11 \times 2 = 2002
91 \times 11 \times 3 = 3003
Write next seven steps. Check, whether the result is correct.
Try the pattern for 143 \times 7 \times 1, 143 \times 7 \times 2 .....



How would we multiply the numbers 13680347, 35702369 and 25692359 with 9 mentally ? What is the pattern that emerges. 



The sum of the series \displaystyle 1+\frac{1.3}{6}+\frac{1.3.5}{6.8}+....\infty is?



If n\epsilon N and n is even, then \dfrac {1}{1.(n - 1)!} + \dfrac {1}{3!(n - 3)!} + \dfrac {1}{5!(n - 5)!} + .... + \dfrac {1}{(n - 1)!1!} equals.



Solve the following equations.
\displaystyle\, 7^{x + 2} - \frac{1}{7}\cdot 7^{x + 1} - 14.7^{x - 1} + 2.7^x = 49



If f\left( x+y,\  x-y \right) =xy, then the arithmetic mean of f(x,y) and f(y,x) is



If \alpha =\dfrac { 5 }{ 2!3 } +\dfrac { 5.7 }{ 3!{ 3 }^{ 2 } } +\dfrac { 5.7.9 }{ 4!{ 3 }^{ 3 } },.... then find the value of { \alpha  }^{ 2 }+4\alpha



Prove 1+2+3+4+5+\cdots+n=\dfrac{n(n+1)}{2}



Find the n^{th} term and the sum of first n terms of the sequence 0.3, 0.33, 0.333,0.3333,.....



Find the sum of the series {1}^{2}+({1}^{2}+{2}^{2})+({1}^{2}+{2}^{2}+{3}^{2})+.....



1 \, + \, \dfrac{3}{2} \, + \, \dfrac{5}{4} \, + \, \dfrac{7}{8} \, + \, ..... \, n \, terms



The natural numbers are written in the form of a triangle as shown below :
The sum of numbers in all the n rows .
1000705_4cfbf2f994424b4fbd668cfefc69a799.png



The sum of 'n' terms of  series \bigg(1-\dfrac{1}{n}\bigg)+\bigg(1-\dfrac{2}{n}\bigg)+\bigg(1-\dfrac{3}{n}\bigg)+........... will be



There are n AM's between 1\,\& 31 such that 7th mean: {\left( {n - 1} \right)^{th}} mean  = 5:9, then find the value of n.



If \displaystyle x=\sum _{ n=0 }^{ \infty  }{ { a }^{ n } } ,y=\sum _{ n=0 }^{ \infty  }{ { b }^{ n } } ,z=\sum _{ n=0 }^{ \infty  }{ { \left( ab \right)  }^{ n } } ,
where a,b<1 then prove that xz+yz=xy+z.
Another form:
For 0<\theta ,\phi -\cfrac { \pi  }{ 2 } if \displaystyle x=\sum _{ n=0 }^{ \infty  }{ \cos ^{ 2n }{ \theta  }  }
\displaystyle y=\sum _{ n=0 }^{ \infty  }{ \sin ^{ 2n }{ \phi  } , } z=\sum _{ n=0 }^{ \infty  }{ \cos ^{ 2n }{ \theta  }  } \sin ^{ 2n }{ \phi  } then prove that xz+yz-z=xy.



If a , b are = ive , then from any integer  n prove that 
(a \, + \, b)^n \, < \, 2^n\, (a^n \, + \, b^n )



Find the 50_{th} term of the sequence:
  \dfrac{1}{n},\,\dfrac{n+1}{n},\,\dfrac{2n+1}{n}...................



(a) If the roots of the equation, (b-c){ x }^{ 2 }+(c-a)x+(a-b)=0 be equal, then prove thar a,b,c are in arithmetical progression.
(b) If a(b-c){ x }^{ 2 }+b(c-a)x+c(a-b)=0 has equal roots, prove that a,b,c are in harmonical progression.



If y = x^n  log x then 
y_n \, = \, n!\left [ log \, \, x \, + \, 1 \, +\dfrac{1}{2}\, +\dfrac{1}{3} .... \, + \, \, \dfrac{1}{n} \right ]
\forall \, n \, \epsilon \, N and x > 0.



A sequence of real numbers a_1, a_2,___ a_n is such that a_1=0, |a_2|=|a_1+1|, |a_3|=|a_2+1|, _______|a_n|=|a_{n-1}+1|. Show that A.M. of these numbers is always greater or equal to -1/2.



Show that \dfrac{1}{a_1a_2}+\dfrac{1}{a_2a_3}+________+\dfrac{1}{a_{n-1}a_n}=\dfrac{n-1}{a_1a_n}.



What is the expression of \tan^3x?



Find the arithmetic mean 
If 69.5 is the mean of 72, 70, x, 62, 50, 71, 90, 64, 58 and 82. Find the value of x.



Sum upto n the series
6+66+666+.....



Observe the following pattern:2^{6} = 64
2^{5} = 32
2^{4}=16
2^{3} =8
2^{2} =4
2^{1} =?
2^{0} =?
You can guess the value of 2^{0} by just studying the pattern!
You find that 2^{0} =1
If you start from 3^{6} = 729, and proceed as shown above finding 3^{5},3^{4},3^{3}... etc, what will be 3^{0} = ?



Find the number of terms in the series 20 + 19 \dfrac{1}{3} + 18 \dfrac{2}{3} + .... of which the sum is 300, explain the double answer.



2+2+4+1+8+\dfrac{1}{2}+16+......+55 term sum =?



Find the sum of series 1+(2)(3)+(4)+(5)(6)+7+.... upto 50 term.



Find the smallest among three consecutive odd natural numbers whose sum is 87.



Find the sum of series
0.5 + 0.55 + 0.555 + ...



The sum of 10^2+11^2+12^2+....+20^2 is 



A plant doubled its height each yer until it reached its maximum height over the course of 12 years How many years did it take for it to reach half of its maximum height? 



If a, b, c be the respective sums of the n terms, next n terms and of G.P. then prove that a, b, c are in G.P.



If \displaystyle \sum _{r=1}^{n}{T_{r}=5n+2} and \displaystyle  \sum _{r=1}^{n}{T_{r}^{2}=an^{3}+bn^{2}+cn+d} then the value of d+b+c



Find the next term 3, 5, 7, 10, 11, 15, 15, 20,__



\left( 1+\dfrac { 3 }{ 1 }  \right) \left( 1+\dfrac { 5 }{ 4 }  \right) \left( 1+\dfrac { 7 }{ 9 }  \right) ...\left( 1+\dfrac { \left( 2x+1 \right)  }{ { n }^{ 2 } }  \right) ={ \left( n+1 \right)  }^{ 2 }.



Prove \left( 1+\dfrac { 1 }{ 1 }  \right) \left( 1+\dfrac { 1 }{ 2 }  \right) \left( 1+\dfrac { 1 }{ 3 }  \right) ...\left( 1+\dfrac { 1 }{ { n } }  \right) ={ \left( n+1 \right)  }.



The arithmetic mean of the nine numbers in the given set {\left\{9, 99, 999, 999999999 \right\}} is a 9 digit numbers N, all whose digits are distinct. The number N does not contain the digit



\dfrac{1}{{1.2.3}} + \dfrac{1}{{2.3.4}} + \dfrac{1}{{3.4.5}} + .... + \dfrac{1}{{m\left( {n + 1} \right)\left( {n + 2} \right)}} = \dfrac{{n\left( {n + 3} \right)}}{{4\left( {n + 1} \right)\left( {n + 2} \right)}}



Sum the series :
2^{1/4} \, \cdot \, 4^{1/8} \, \cdot \, 8^{1/16} \, \cdot \, 16^{1/32} \, .... is equal 



Let a_1, a_2, a_3, a_4, a_5 be a G.P. of positive real numbers such that the A.M. of a_2 and a_4 is 117 and the G.M. of a_2 and a_4 is 108. Then the A.M. of a_1 and a_5 is:



 In the given squences 2,5,6.....50and 3,5,7.....60
How many terms are common to both?



A series of numbers is given as 1, \dfrac{1}{2}, -4, 4, \dfrac{1}{8}, -16, ...... The next number in the series is



Find the sum of place value of last 3 places in the sum of first '10' terms of : 4 + 44 + 444 + ...



Find the value of \lim\limits_{n \to \infty}\dfrac{1}{n}\displaystyle \sum^{n}_{r=1}\sin^2 \dfrac{r\pi}{n}.



Sum: \frac{3}{1^2.2^2}+\frac{5}{2^2.3^2}+\frac{7}{3^2.4^2}+....... to n terms.



Vijay is planting trees. He has enough trees to plant 6, 7 or 14 trees in each row. What is the least number of trees Vijay could have?



1+3+5+.........+2n-1=



Find the n^{th} term and sum to n terms of the following series:
2 + 6 + 12 + 20 +____



If the roots of x^{3}+px^{2}+qx+r=0 are in G.P., find the relation between p,q,r.



Find the n^{th} term and sum to n terms of the following series:
3 + 6 + 11 + 18 +____



The value of \sum_\limits{i=1}^n\sum_\limits{j=1}^i\sum_\limits{k=1}^j 1=



cos \{\frac{2\pi}{2^{64}-1}\}cos\{\frac{2^2\pi}{2^{64}-1}\}......cos\{\frac{2^{64}\pi}{2^{64}-1}\}=



find the sum : tan x tan 2x + tan2x tan 3x + .............+ tan nx   tan(n+1)x .



If statement P(n) is "3n + 1 is even". Then verify that statement P(1) is true but P(2) is not true.



7, 13, 19,...., 205.



Prove that \dfrac{(2n!)}{n!}=2^n (1.3.5....(2n-1)).



Find the n^{th} term and sum to n terms of the following series:
1 + 9 + 24 + 46 + 75 +_____



Find the period of f\left( x \right) =\tan { \left( x+4x+9x+.......+{ n }^{ 2 }x \right)  }



Prove that { 2 }^{ 1/4 }\cdot { 4 }^{ 1/8 }\cdot { 8 }^{ 1/16 }\cdot { 16 }^{ 1/32 }\cdot ........\cdot \infty =2



1+2+3+....+n < \dfrac{1}{8} (2n+1)^2.



5+3+2=151022
9+2+4=183652
8+6+3=482466
5+4+5=202541
Then . . .
7+2+5= ?????



Find the total number of ways in which six '+' and four '-' signs can be arranged in a line such that no two '-' signs occur together.



Find the sum to the first n terms of the series : 
\frac{1}{{(1 - x)(1 - {x^3})}} + \frac{{{x^2}}}{{(1 - {x^3})(1 - {x^5})}} + \frac{{{x^4}}}{{(1 - {x^5})(1 - {x^7})}} + ...,x \ne 1, - 1,0



Find the {8^{th}} term of the sequence whose first three terms are 3,\, 3,\, 6 and each term after the second is the sum of three two terms preceding it .



Solve: \cfrac{1}{2}+\cfrac{1}{6}+\cfrac{1}{12}+\cfrac{1}{20}+.....+\cfrac{1}{1892}+\cfrac{1}{1980}=?



If a, b, c , d are in G.P., show that 
i) a^2 + b^2 , \, b^2 + c^2, \, c^2 + d^2 are in G.P



Find m if the following equation holds true:  1.2 + 4.3 + 9.4 + ... n^2 (n + 1) = \dfrac{n}{m} (n + 1) (n + 2) (3n + 1)



If \frac{1}{{1!9!}} + \frac{1}{{3!7!}} + \frac{1}{{5!5!}} + \frac{1}{{9!1!}} = \frac{{{2^n}}}{{10!}}
Then n =



In a sequence of 21 terms, the first 11 term are in A.P with common difference 2 and the last 11 terms are in G.P. with common ratio 2. If the middle term of the A.P. is equal to the middle term of  G.P. The find the middle term of the entire sequence.



Find n^{th} term of the series
3\times 1^{2} , 5\times 2^{2} , 7\times 3^{2} , .....



Prove : {1^2} + \left( {{1^2} + {2^2}} \right) + \left( {{1^2} + {2^2} + {3^2}} \right) +  \ldots upto n terms = \dfrac{{n{{\left( {n + 1} \right)}^2}\left( {n + 2} \right)}}{{12}}



1^2+\left(\dfrac{3}{2}\right)^2+2^2+\left(\dfrac{5}{2}\right)^2+... to 40 terms.



Find the average of first 40 natural numbers? 



Find the sum to n terms of the series
05+11+19+29+......?



Find the sum of the series \displaystyle\sum^n_{r=1}r\times r!



If sum to infinity of the series 3-5r+7r^2-9r^3+.... is \dfrac{14}{9}, find r.



A snail starts moving towards a point 3cm away at a pace of 1cm per hour. As it gets tired, it covers only half the distance compared to previous hour in each succeeding hour. In how much time will the snail reach his target?



Find the nth term and the sum of n terms of the series 1.2.4+2.3.5+3.4.6+....



The sum of the nth term of the series 1.2.5 + 2.3.6 + 3.4.7 + .....n terms is



Find the sum of 5+7+9+11+13+.... for 50 terms.



If the roots of the equation x^3-12x^2+39x-28=0 are in A.P., then their common difference will be,



 Find the sum of the given series 6+11+16+21+......+86 



Find:1^3-2^3+3^3-4^3+.....+9^3=?



Sum to infinite terms the following series:
1 + 4x + 7x^{2} + 10x^{3} + ..., |x| < 1.



Sum to n terms the following series:
1 + 2x + 3x^{2} + 4x^{3} + ...,|x| < 1



Sum to n terms of the following series:
1+\left(1+\dfrac{1}{2}\right) + \left(1 + \dfrac{1}{2} + \dfrac{1}{4}\right) + ,...



Sum to n terms the series :
1 \times 2+2\times 3+3\times 4+4\times 5+..



Let d be the minimum value of f(x)=5x^2-2x+\dfrac{26}{5} and f(x) is symmetric about x=r. If \sum_{n=1}^{\infty}(1+(n-1)d)r^{n-1} equals \dfrac{p}{q}, where p and q are relative prime, then find the value of (3q-p).



Sum to n terms the series :
1\times 3+3\times 5+5\times 7+7\times 9+..



A binary string of length N is a sequence of 0s and 1s. For example, 01010 is a binary string of length 5. For a string A we write A_{i} to refer to the letter at the i^{th} position.



Find missing.....
1162888_fe1f31ba7b334915b9e1fc5336f5f9c8.png



Find the sum of 1st n terms of the series 5+7+13+85+.



If sum to infinity of series 3-5r+7r^{2}-9r{3}+. is 14/9, find r.



If a,b,c are positive real numbers in A.P and {a^2},{b^2},{c^2} are in H.P, then\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} =



If a,b,c are in A.P,  then show that 10^{ax+10},10^{bx+10},10^{cx+10},x \neq 0 are in G.P



Sum of the series 4+6+9+13+18+......n terms , is 



Find the {10^{th}} form of the series 3 + 6 + 12 + ....



The sum of the series 1 + \frac{1}{6} + \frac{1}{{18}} + \frac{7}{{324}} + ....\infty , is 



Find the average of first 20 multiples of 7



{\alpha _r},{\beta _r}({\alpha _r} < {\beta _r})\;{\text{are}}\;{\text{the}}\;{\text{root}}\;{\text{of}}\;{x^2} - {r^2}\left( {r + 1} \right)x + {r^5} = 0 Find the value of \sum\limits_{r = 1}^n {\left( {3{\alpha _r} + 2{\beta _r}} \right): - }



Find the sum to n terms of the series, whose nth term is given by {n}^{2}+{2}^{n}.



2\cdot 3+3\cdot 4+4\cdot 5+..... upto n terms Find sum of n of series.



{1^3} - {2^3} + {3^3} - {4^3} + .... + {9^3} =



Solve 1 - \dfrac { 1 } { 2 } + \dfrac { 1 } { 3 } - \dfrac { 1 } { 4 } + \dots  \infty 



Find n if the sum of the first n terms of the series \sqrt{3}+\sqrt{75}+\sqrt{243}+\sqrt{507}+......is435\sqrt{3}.



Show that \left( 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 } + \ldots \ldots . \infty \right) ^ { 3 / 2 } = 1 + 3 x + 6 x ^ { 2 } + 10 x ^ { 3 } + \ldots \ldots , \infty , where | x | < 1



The sum of  series \frac{1}{3\times 6}+\frac{1}{6\times 9}+\frac{1}{9\times 12}+...... is:



The traffic lights at three different road crossing change after every 48 seconds, 72 seconds and 108 seconds respectively.If they change simultaneously at 7 : 00 a.m, at what time will they change simultaneously again? 



satisfies the recurrence relation b_{n+1}=\frac{1}{3}\left ( 2b_{n}+\frac{125}{b^{2}_{n}} \right ),b_{n}\neq 0



\sqrt{1+3+5+7+...}.



As income is 10\% more than that of B. How much per cent is the income of B less than that of A?



An Arithmetic Series is defined as 
f\left( n \right) = a + \left( {n - 1} \right)d,n \in N, prove that A.M. of f\left( 1 \right) and f\left( {2n - 1} \right) is f\left( n \right).



What comes next in the series?
2,3,6,10,17,28,?



Find the co-efficient of x ^ { 7 } in the series of e ^ { 2 x }.



Show that : \dfrac{1^3}{1} + \dfrac{1^3 + 2^3}{1+3} + \dfrac{1^3 + 2^3 + 3^3}{1+3+5} .........up to n terms = \dfrac{n}{24}[2n^2 +9n +13]



Find the sum of  8 + 88 + 888 + 8888 + \cdots



Solve
1^{3}+\dfrac{1^{3}+2^{3}}{2}+\dfrac{1^{3}+2^{3}+3^{3}}{3}+...



Find the A.M. of the series 1, 2, 4, 8, 16, .......2^{n}



Find the sum to infinity of the series  1 + \dfrac { 2 } { 3 } + \dfrac { 6 } { 3 ^ { 2 } } + \dfrac { 10 } { 3 ^ { 3 } } + \dfrac { 14 } { 3 ^ { 4 } } + \ldots



Evaluate \sum _ { k = 1 } ^ { n } \left( 2 ^ { k } + 3 ^ { k - 1 } \right)



\frac { 1 } { 1 \times 2 } + \frac { 1 } { 2 \times 3 } + \frac { 1 } { 3 \times 4 } + \ldots what will be the n^{th} term?



Sum to infinite terms of 
1 + 3 x + 5 x ^ { 2 } + 7 x ^ { 3 } + \ldots \ldots + \infty ( \boldsymbol { H } \boldsymbol { f } | x | < 1 ) is 



Write the next term of each of the following sequences :
0, 2, 6, 12, 20,.....



The formula of the sum of first n natural numbers is S = \frac { n ( n + 1 ) } { 2 }. If the sum of first n natural number is 325, find n



The decimal expansion of the rational number \dfrac{43}{2^{4}\times 5^{3}} will terminate after how many places of decimal.



Find the sum of the series : \dfrac{1}{3^{2}+1}+\dfrac{1}{4^{2}+2}+\dfrac{1}{5^{2}+3}+\dfrac{1}{6^{2}+4}+...\infty



Write the next term of each of the following sequences :
6, 9, 16, 27, 42,.....



Can you find the missing numbers?
1274972_fcfffba8b0ba474390de044616782866.png



Find the sum of first 1000 natural numbers and first 1000 non-positive integers.



Complete the addition and subtraction box.
1272210_dedbfaecf4f74dbdb50555323925a021.png



Evaluate:
1 \dfrac { 1 } { 2 } + 1 \dfrac { 1 } { 3 } + 1 \frac { 1 } { 4 } + \ldots + 1 \dfrac { 1 } { 19 } =



Assume that a,b,c and dare positive integers such that a^{5}=b^{4},c^3=d^{2} and c-a=19. Determine d-b



1+2x+3x^2+4x^3+.....\infty; |x| < 1.



Prove that 
{1}^{2}+{3}^{2}+{5}^{2}+...+{\left(2n-1\right)}^{2}=\dfrac{n}{3}\left(2n-1\right)\left(2n+1\right)



Find the value of \dfrac{50}{72}+\dfrac{50}{90}+\dfrac{50}{110}+\dfrac{50}{132}+...+\dfrac{50}{9900}.



\dfrac{a}{1+i}+\dfrac{a}{(1+i)^{2}}\dfrac{a}{(1+i)^{3}}+....+\dfrac{a}{(1+i)^{n}}



In the four numbers first three are in G.P. and last three are in A.P. whose common difference isIf the first and last numbers are same, then first will be



What will be in the '?' mark
1306899_1c49009ecda04a1ca08a1af21d9ec6dc.png



Prove that { 3 }^{ \frac { 1 }{ 2 }  }\times { 3 }^{ \frac { 1 }{ 4 }  }\times { 3 }^{ \frac { 1 }{ 8 }  }.....=3



Given that  \left( 1 + x + x ^ { 2 } \right) ^ { n } = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \ldots + a _ { 2 n } x ^ { 2 n } ,  find the values of
(i) a _ { 0 } + a _ { 1 } + a _ { 2 } + \dots . + a _ { 2 n }
(ii) a _ { 0 } - a _ { 1 } + a _ { 2 } - a _ { 3 } \dots . . + a _ { 2 n }



What is the {1025}^{th} term of the sequence 1,2,2,4,4,4,4,8,8,8,8,8,8,8,8,?



Find the sum of first n terms of the series {1^2} + \left( {{1^2} + {2^2}} \right) + \left( {{1^2} + {2^2} + {3^2}} \right) + .....



Write the seventh term of the series \dfrac{1}{\sqrt{2}}, -2, \dfrac{8}{\sqrt{2}},\dots \dots



Solve:
2\dfrac {1}{5}+3\dfrac {1}{7}



Find the sum of 1+4+7+10+....... to 22 terms.



Find the mean weight from the following table.
Weight (kg)293031
3233
No.of children2001040305



How many terms of the geometric progression 1+4+16+64+.......... must be added to get sum equal to 5641?



Find the value of y if 1+4+7+10+.....+y=287



Prove that \dfrac{1}{1+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+ ............. +\dfrac{1}{\sqrt{8}+\sqrt{9}}=2



If a boy bats 100 times and gets 40 hits, what is his batting average ?



Fill this square using all numbers from 21 to 29.
1379370_17c1363367ef4ffb974c03df66f0e92f.png



look at the pattern and Solve
1            =  1X1
121        =__x__
12321    =__x__
1234321=__x__
_____    =11111x11111



If the mean of 5 observations x,x+2,x+4,x+6 and x+8 is 11, find the value of x.



In a question, 55+10 = 20 and 45+40 = 49, then 95+70 = ?



Calculate the arithmetic mean, range, median and mode of the given data:
2,\,4,\,7,\,4,\,9,\,5,\,7,\,3,\,6,\,7



Fill in the blanks in the expressions with the proper numbers.
(1\ \times 7)=(\Box  \times 1)



Find the mean been the following:-
2,2,0,4,3,9,5,7



x=7 when y=5. Find y when x=14.



if 1 + {x^2} = \sqrt 3 x,\,\,then,\,\,\prod\limits_{n = 1}^{24} {\left( {{x^n} + \dfrac{1}{{{x^n}}}} \right)} is equal to:



Which month has the maximum number of birthdays?_____________
1396361_0b579ff450c34c508a7e6b97d0a683cd.png



Find the mean of the following.
First seven multiples of 6



Find the mean of the following
First eight even natural numbers.



If {S_n} denotes the sum of n terms of an AP whose common differences is d, show that d = {S_n} - 2{S_{n - 1}} + {S_{n - 2}}



The mean of 12,65,84,75,a is 50 Find a



The value of p if ,the mean of 23,25,p is 24



Find the missing number:
\begin{array}{|c|c|c|}\hline 5 & {6} & {12} \\ \hline 4 & {3} & {4} \\ \hline 2 & {3} & {?} \\ \hline 18 & {27} & {96} \\ \hline\end{array}



Observe the given pattern
4\times 0+1=01
4\times 1+2=06
4\times 2+3=11
4\times3+4 =16 and find the {15}^{th} term.



If the mean of x+4,5,3,6,7 is 5 then find x



Solve:
\frac{1}{{2.5}} + \frac{1}{{5.8}} + \frac{1}{{8.11}} +  -  -  -  + \frac{1}{{(3n - 1)(3n + 2)}}



A student scored marks 46 % in English, 67 % in Maths, 53 % in Hindi, 72 % in History and 58 % in Economics. As compared to other subject weightage given to Mathematics, then find weightage mean marks of students.



Four terms are in A.P. If sum of numbers is 50 and largest number is four times the smaller one, then find the terms.



Find the sum of the series \displaystyle  \frac { 1 }{ 1.2 } -\frac { 1 }{ 2.3 } +\frac { 1 }{ 3.4 } -.....\infty



Simplify : 15\times \left( -25 \right) \times \left( -4 \right) \times \left( -10 \right)



1^{3}+2^{3}+3^{3}+.....+n^{3}



The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.



The seventh term of a G.P. is 8 times the fourth term and 5th term isFind the G.P.



Find AM of 12 and 14.



Find AM of 15 and -13.



Find the sum of the series whose nth term is :
n^{3}-3^{n}



Find AM of 7 and 27.



Find the sum of the series whose nth term is:
n(n+1)(n+4)



Find the arithmetic mean between 9 and 19.



Find the sum of the series whose nth term is:
(2n-1)^{2}



Find the sum of the series whose nth term is:
2n^{3}+3n^{2}-1



Find the sum of the following series to n terms:
1.2.4+2.3.7+3.4.10+...



Find the 20^{th} term and the sum of 20 terms of the series : 2\times 4+4\times 6+6\times 8+...



An insect is on 0 point of a number line, hopping towards 1. She covers half the distance from her current location to 1 with each hop. So, she will be at 1/2 after one hop, 3/4 after two hops, and so on.
Where will the insect be after n hops?
1792969_2e62eda7200047a9b1c5125bc5f1fa0b.png



Which of the following form of an A.P.? Justify your answer.
-1, -1, -1, -1, .......



Justify whether it is true to say that -1, \dfrac{-3}{2}, -2, \dfrac{5}{2}, ..... forms an A.P. as a_2-a_1= a_3-a_2.



The time taken by Rohan in five different races to run a distance of 500 m was 3.20 minutes, 3.37 minutes, 3.29 minutes, 3.17 minutes and 3.32 minutes. Find the average time taken by him in the races.



Find the arithmetic mean between 15 and -7.



Find the arithmetic mean between -16 and -8.



Find the geometric progression whose 4^{\text {th }} term is 54 and the 7^{\text {th }} term is 1458.



In which of the following situations do the list of numbers involved form an A.P.?
Give reasons for your answers.
The number of bacteria in a certain food item after each second, when they double in every second.



Let S denote sum of the series \dfrac{3}{2^3} + \dfrac{4}{2^3} + \dfrac{5}{2^3} + \dfrac{6}{2^5} + ... \infty. Then the value of S^{-1} is.



The sum of first n natural numbers is given by
\cfrac { 1 }{ 2 } {n}^{2}+\cfrac { 1 }{ 2 } n. Find
 The sum of first 5 natural numbers



Fourth and seventh terms of a G.P. are \dfrac{1}{18} and \dfrac{-1}{486} respectively. Find the G.P.



Define\space geometric progression.



If n arithmetic mean are inserted in between 1 and 51 such that ratio of 4th and 7th arithmetic mean is 3:5 then find the value of n.



Find the G.P. \dfrac{1}{27} , \dfrac{1}{9} , \dfrac{1}{3} , ............. 81 ; find the product of fourth term from the beginning and the fourth term from the end.



Simplify 1+\dfrac{1}{3}.\dfrac{1}{4}+\dfrac{1.4}{3.6}.\dfrac{1}{4^2}+...



Find the arithmetic mean of: 
(m + n)^2 and (m n)^2 



Write the formula to find arithmetic mean of 3 numbers



Find the arithmetic mean of: 
3x 2y and 3x + 2y



Find the arithmetic mean of: 
-5 and 41



the mean \bar{x}



Prove that :
\sqrt{2}=1+\dfrac{1}{2^2}+\dfrac{1.3}{2! . 2^4}+\dfrac{1.3.5}{3! . 2^6}+...



Simplify 1+\dfrac{1}{10}+\dfrac{1.4}{10.20}+\dfrac{1.4.7}{10.20.30}+...



Prove that:
\dfrac{5\sqrt{2}}{7}=1+\dfrac{1}{10^2}+\dfrac{1.3}{1.2}\dfrac{1}{10^4}+...



1+\dfrac{1}{4}+\dfrac{1.4}{4.8}+\dfrac{1.4.7}{4.8.12}+...



Simplify 1-\dfrac{1}{2}.\dfrac{1}{2}+\dfrac{1.3}{2.4}\bigg(\dfrac{1}{2}\bigg)^2-\dfrac{1.3.5}{2.4.6}\bigg(\dfrac{1}{2}\bigg)^3+...



Prove that:
\bigg(\dfrac{3}{2}\bigg)^{1/3}=1+\dfrac{1}{3^2}+\dfrac{1.4}{1.2}.\dfrac{1}{3^4}+\dfrac{1.4.7}{1.2.3}.\dfrac{1}{3^6}+...



Prove that:
(1+x)^n=2^n\Bigg[1-\dfrac{n(1-x)}{(1+x)}+\dfrac{n(n+1)}{2!}\Bigg(\dfrac{1-x}{1+x}\Bigg)^2-\,...\Bigg]



Evaluate S=1+\dfrac{4}{5}+\dfrac{7}{5^2}+\dfrac{10}{5^3}+...... to infinite terms. Find 16S



Find the sum of n terms of { 1 }^{ 2 }+\left( { 1 }^{ 2 }+{ 2 }^{ 2 } \right) +\left( { 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 } \right) +\left( { 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+{ 4 }^{ 2 } \right) +... from that find the sum of the first 10 terms



What is the value of ({2^2+ 4^2+ 6^2 +... + 20^2})-({1^2 + 3^2 + 5^2+ ... + 19^2})?



Show that the 3{ n }^{ th } convergent to
    \dfrac { 1 }{ 5- } \dfrac { 1 }{ 2- } \dfrac { 1 }{ 1- } \dfrac { 1 }{ 5- } \dfrac { 1 }{ 2- } \dfrac { 1 }{ 1- } \dfrac { 1 }{ 5- } \cdots is \dfrac { n }{ 3n+1 } .



Find the sum of \displaystyle\frac{1}{(1+x)(1+ax)}+\frac{a}{(1+ax)(1+a^2x)}+\frac{a^2}{(1+a^2x)(1+a^3x)}+.... to n terms.



If x=a+\displaystyle\frac{1}{b+}\frac{1}{b+}\frac{1}{a+}\frac{1}{a+}....,
y=b+\displaystyle \frac{1}{a+}\frac{1}{a+}\frac{1}{b+}\frac{1}{b+}....,.
Then show that (ab^2+a+b)x-(a^2b+a+b)y=a^2-b^2.



Sum the following series to n terms and to infinity \displaystyle\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+.....



Prove that \cos^3 \theta + \cos^3 \left( \theta - \dfrac {2 \pi}{n} \right) + \cos^3 \left( \theta - \dfrac {4 \pi}{n} \right) + ... to n terms = 0



Find Sum of Series \tan^2 x \tan 2x + \dfrac {1}{2} \tan^2 2x \tan 4x + \dfrac {1}{2^2} 4x \tan 8x + .... to n terms.



Find the sum to infinite terms of the series \dfrac {x}{1 - x^{2}} + \dfrac {x^{2}}{1 - x^{4}} + \dfrac {x^{4}}{1 - x^{8}} + .....



Find the sum of the following infinite series:
\displaystyle\sum _{ n=0 }^{ \infty  }{ \cfrac { 1 }{ n! } \left[ \sum _{ k=0 }^{ n }{ \left( k+1 \right) \int _{ 0 }^{ 1 }{ { 2 }^{ -\left( k+1 \right) x }dx }  }  \right]  } 



Find \ \sin\theta+\ \sin({\pi}+\theta) +\ \sin(2\pi+\theta) +\ \sin(3\pi+\theta)+ \cdots upto 2021 terms.



What is value of 
1 + x + {x^2} + {x^3} + {x^4} + ..... 
where x \ne 1



1.3+3.5+5.7+....+\left( {2n - 1} \right)\left( {2n + 1} \right) = \frac{{n\left( {4{n^2} + 6n - 1} \right)}}{3}



\cos { \dfrac { x }{ 2 }  } \times \cos { \dfrac { x }{ { 2 }^{ 2 } }  } \times \cos { \dfrac { x }{ { 2 }^{ 3 } }  } ..........\times \cos { \dfrac { x }{ { 2 }^{ n } }  }A\rightarrow \infty\dfrac { \sin { { 2 }^{ n } } A }{ { 2 }^{ n }\sin { A }  }



If S=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+............. to \infty then find \frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+............. to \infty in terms of S.



Find the sum of infinite terms of following series.
\dfrac{3}{7}+\dfrac{5}{21}+\dfrac{7}{63}+\dfrac{9}{189}+.....



Sum of infinity the following series:
1 + 4x^{2} + 7x^{4} + 10x^{6} + ..., |x| < 1.



Find the sum to the series 1.n+2(n-1)+3(n-2)+...+n.1



Sum to infinite terms the following series:
1 + 5x^{2} + 9x^{4} + 13x^{6} + ...., |x| < 1



If S_n = \dfrac{3}{4} + \dfrac{5}{36} + \dfrac{7}{144} + \dfrac{9}{400} +... to n terms, then find \dfrac{1}{1 - S_{40}}.



Sum the following series
\dfrac{1}{1.4.7}+\dfrac{1}{4.7.10}+\dfrac{1}{7.10.13}+.... to n terms



Sum to infinite terms the following series:
1 + 3x + 5x^{2} + 7x^{3} + ...., |x| < 1.



In the given diagram the circle stands for "educated", square for "lazy", triangle for "urban" and the rectangle for "honest" people. The different regions in the diagram are numbered from 2 to 13. Number 4 will show which of the case?
1183594_7a93d1c19a9f4ff2853776812639478e.png



 If  \left| x \right| < 1 and  \left| y \right| < 1, find the sum to infinity to the series 

                                   \left( {x + y} \right) + \left( {{x^2} + xy + y} \right) + \left( {{x^3} + {x^2}y + x{y^2} + {y^2}} \right)



If a,\ b,\ c are in A.P. and P is the A.M. between a and b, and q is the A.M. between b and c, show that b is the A.M. between p and q.



Let x = 111 .... 11 (20 digits)
      y = 333 .... 33 (10 digits)
and  z = 222 .... 22 (10 digits),
The value of \dfrac{x - y^2}{z}.



Show that \cfrac { 1.2 ^{ 2 }+{ 3 }^{ 2 }+...+n }{ { 1 }^{ 2 }. 2+{ 2 }^{ 2 }. 3+...+ n  } =\cfrac { 3n+5 }{ 3n+1 }



If {\left(1+x-2{x}^{2}\right)}^{20}=\sum_{r=0}^{40}{{a}_{r}{x}^{r}}  then the value of {a}_{1}+{a}_{3}+{a}_{5}+...+{a}_{39}



What is the ninth term of the sequence whose general term a_{n}=(-1)^{n-1}n^{3}.



The sum of the infinite series   \dfrac { 1.3 } { 2 } + \dfrac { 3.5 } { 2 ^ { 2 } } + \dfrac { 5.7 } { 2 ^ { 3 } } + \dfrac { 7.9 } { 2 ^ { 4 } } + \ldots . . . . . . . \infty.



If \dfrac{{a}^{n}+{b}^{n}}{{a}^{n-1}+{b}^{n-1}} is the A.M. between a and b, then find the value of n



Find the sum of the series 
\dfrac{3 \cdot 5}{5 \cdot 10} + \dfrac{3 \cdot 5 \cdot 7}{5 \cdot 10 \cdot 15} + \dfrac{3 \cdot 5 \cdot 7 \cdot 9}{ 5 \cdot 10 \cdot 15 \cdot 20} + ..... \infty.



The average of two numbers 'a' and 'b' is 68 and the average of 'b' 'c' is 70 and that and 'c' isFind the values of a, b, c.



Find the mean of first five prime numbers.



The sum of the series \dfrac{1}{1 ! (n-1) ! }+\dfrac{1}{ 3! (n-3) !}+\dfrac{1}{5 ! (n-5)!}+.......+\dfrac{1}{(n-1) ! 1!}



Class 11 Engineering Maths Extra Questions