CBSE Questions for Class 11 Engineering Maths Sequences And Series Quiz 10 - MCQExams.com

The sum of $$\displaystyle \sum^{\infty}_{n=1}\tan^{-1}\left(\dfrac {2}{n^{2}+n+4}\right)$$ is equal to-
  • $$\tan^{-1}2$$
  • $$\dfrac {\pi}{2}+\tan^{-1}2$$
  • $$\dfrac {\pi}{2}-\tan^{-1}2$$
  • $$\dfrac {\pi}{4}$$
$$496 : 204 : : 329 : ?$$
  • $$90$$
  • $$110$$
  • $$115$$
  • $$135$$
If $${ sin }^{ 4 }\alpha .{ cos }^{ 2 }\alpha =\sum _{ r=0 }^{ 3 }{ { C }_{ r } } .cos(2r\alpha )$$ then $${ C }_{ 0 }+{ C }_{ 1 }+{ C }_{ 2 }+{ C }_{ 3 }=$$
  • 0
  • 3
  • 4
  • 7
If $$S_n=\dfrac{7}{4.1.2}+\dfrac{10}{4^2.2.3}+\dfrac{13}{4^3.3.4}+....$$ then $$S_{\propto}$$ is equal to?
  • $$\dfrac{5}{2}$$
  • $$\dfrac{9}{8}$$
  • $$\dfrac{3}{2}$$
  • $$1$$
Find the missing number.
1306969_c830724f37e04dcebf1e546f99ec639f.png
  • $$74$$
  • $$62$$
  • $$91$$
  • $$97$$
The sum of the following series $$1+6+\dfrac{9(1^2+2^2+3^2)}{7}+\dfrac{12(1^2+2^2+3^2+4^2)}{9}+\dfrac{15(1^2+2^2+....+5^2)}{11}+.....$$ up to $$15$$ terms is:
  • $$7820$$
  • $$7830$$
  • $$7520$$
  • $$7510$$
The value of the series $$\dfrac{1}{3!}+\dfrac{2}{5!}+\dfrac{3}{7!}+..... $$  to $$\infty $$   is equal to 
  • $$\dfrac{1}{2}e$$
  • $$\dfrac{1}{2e}$$
  • $$\dfrac{3}{2e}$$
  • $$\dfrac{4}{5e}$$
The sum of the series
  $$\dfrac { 9 }{ 5^{ { 2 } }\cdot 2.1 } +\dfrac { 13 }{ 5^{ { 3 } }\cdot 3.2 } +\dfrac { 17 }{ 5^{ { 4 } }\cdot 4.3 } ..........$$  Infinite terms
  • $$\dfrac { 1 } { 25 }$$
  • $$\dfrac { 1 } { 5 }$$
  • Not a finite number
  • $$\dfrac { 9 } { 5 }$$
The sum to infinity of the series $$1+\dfrac {2}{3}+\dfrac { 6 }{ { 3 }^{ 2 } } +\dfrac { 10 }{ { 3 }^{ 3 } } +\dfrac { 14 }{ { 3 }^{ 4 } }+.... $$ is
  • $$3$$
  • $$4$$
  • $$6$$
  • $$2$$
If $$x_{1},x_{2},....$$ are in H.P and $$x_{1},2,x_{20}$$ are in G.P, then $$\displaystyle \sum _{ 1 }^{ 19 }{ x_{r} .x_{r+1}}=$$ 
  • $$76$$
  • $$80$$
  • $$84$$
  • $$70$$
The value of $$\dfrac { 1 }{ 4 } \tan\dfrac { \pi  }{ 8 } +\dfrac { 1 }{ 8 } \tan\dfrac { \pi  }{ 16 } +\dfrac { 1 }{ 16 } \tan\dfrac { \pi  }{ 32 } +......\infty $$ terms is equal to -
  • $$\dfrac { 5 }{ \pi } -\cfrac { 1 }{ 2 } $$
  • $$\cfrac { 3 }{ \pi } +\cfrac { 1 }{ 2 } $$
  • $$\cfrac { 2}{ \pi } -\cfrac { 1 }{ 2 } $$
  • $$\cfrac { 4 }{ \pi } -\cfrac { 1 }{ 4 } $$
Find the missing number
1312792_39895e31d00e4a2a83fc35377622c5be.png
  • 74
  • 62
  • 91
  • 97
If $$an =\displaystyle \sum_{r =0}^{n} \dfrac{1}{^{n}C_{r}}$$ then $$\displaystyle \sum_{r=0}^{n} \dfrac{r}{^{n}C_{r}}$$ equals
  • $$(n-1)a_{n}$$
  • $$na_{n}$$
  • $$\dfrac{1}{2}na_{n}$$
  • $$None\ of\ these$$
Find the sum of first $$15$$ terms of the sequence whose $${n}^{th}$$ term is $$3+4n$$.
  • $$525$$
  • $$425$$
  • $$495$$
  • $$None\,of\,thes$$
If in a series $$t_n=\dfrac{n+1}{(n+2)!}$$ then $$\displaystyle\sum^{10}_{n=0}t_n$$ is equal to?
  • $$1-\dfrac{1}{10!}$$
  • $$1-\dfrac{1}{11!}$$
  • $$1-\dfrac{1}{12!}$$
  • None of these
Let $$\sum _{ r=1 }^{ n }{ { r }^{ 4 } } =f\left( n \right) ,$$ then $$\sum _{ r=1 }^{ n }{ { \left( 2r-1 \right)  }^{ 4 } } $$ is equal to
  • $$f\left( 2n \right) -16f\left( n \right) $$
  • $$f\left( 2n \right) +7f\left( n \right) $$
  • $$f\left( 2n+1 \right) 8f\left( n \right) $$
  • none of these
If the sum of the first 15 terms of the series $$\lgroup \frac{3}{4} \rgroup^3 + \lgroup 1\frac{1}{2} \rgroup^3 + \lgroup 2\frac{1}{4} \rgroup^3 + 3^3 + \lgroup 3\frac{3}{4} \rgroup^3 + ...........$$ is equal to 225 k. then k is equal to:
  • 9
  • 27
  • 108
  • 54
$$C_1 + 2C_2 + 3C_3 + ...... + nC_n$$ is equal to
  • $$2^{n-1}$$
  • $$2^{n +1}$$
  • $$n. 2^{n -1}$$
  • $$n. 2^{n +1}$$
insert the missing number:  5,8,12,17,23,___,38 
  • $$29$$
  • $$30$$
  • $$32$$
  • $$25$$
The sum of the series $$1+\dfrac{\log_e x}{1!}+\dfrac{(\log_e x)^2}{2!}+........$$ is 
  • $$x$$
  • $$x^2$$
  • $$x^3$$
  • none of these
If x<1, then $$\displaystyle \frac { 1 }{ 1+x } +\frac { 2x }{ 1+{ x }^{ 2 } } +\frac { { 4x }^{ 3 } }{ 1+{ x }^{ 4 } } +.........\infty =$$
  • $$x$$
  • $$\frac { 1 }{ 1+x } $$
  • $$\frac { 1 }{ 1-x } $$
  • $$\frac { 1 }{ x } $$
The sum of the series 1+2(1 +1/n)+ 3$${ \left( 1+1/n \right)  }^{ 2 }+....\infty \quad is\quad given\quad by$$
  • $${ n }^{ 2 }\quad +\quad 1$$
  • $$n\left( n+1 \right) $$
  • $$n{ (1+1/n) }^{ 2 }$$
  • $${ n }^{ 2 }$$
The sum of the series $$ 1 + \frac { 1 + 2 } { 2 } + \frac { 1 + 2 + 3 } { 3 } + \ldots $$ to n terms is
  • $$

    \frac { n ( n + 1 ) } { 2 }

    $$
  • $$

    \frac { n ( n + 3 ) } { 4 }

    $$
  • $$

    \frac { n ( n + 1 ) + n } { 4 }

    $$
  • $$

    \frac { n ( n - 1 ) } { 2 }

    $$
If $$\sum _{ k=2 }^{ n }{ cos{  }^{ -1 }(\frac { 1+\sqrt { (k-1)(k+2)(k+1)k }  }{ k(k+1) }  } )=\frac { 120\pi  }{ \lambda  }$$,then
  • number of even divisors of $$\lambda$$ is 24
  • Sum of proper divisors of $$\lambda$$ is 2418.
  • Sum of proper divisors of $$\lambda$$ is 197.
  • number of ways in which $$\lambda$$ can be expressed as product of two co-prime factors is 4
What is the next number in the series $$2,12,36,80,150?$$
  • $$194$$
  • $$210$$
  • $$252$$
  • $$258$$
The sum of the first n terms of the series $${ 1 }^{ 2 }+2.{ 2 }^{ 2 }+{ 3 }^{ 2 }+2.{ 4 }^{ 2 }+{ 5 }^{ 2 }+2.{ 6 }^{ 2 }+...$$ is $$\dfrac { n{ \left( n+1 \right)  }^{ 2 } }{ 2 } $$ when n is even. When n is odd the sum is
  • $$\dfrac { 3n\left( n+1 \right) }{ 2 } $$
  • $$\dfrac { { n }^{ 2 }\left( n+1 \right) }{ 2 } $$
  • $$\dfrac { { n\left( n+1 \right) }^{ 2 } }{ 4 } $$
  • $${ \left[ \dfrac { n\left( n+1 \right) }{ 2 } \right] }^{ 2 }$$
Value of $$1+\dfrac{1^3+2^3}{1+2}+\dfrac{1^3+2^3+3^3}{1+2+3}+...+\dfrac{1^3+2^3+......15^3}{1+2+.....+15}-\dfrac{1}{2}(1+2+.....+15)$$ is?
  • $$840$$
  • $$720$$
  • $$680$$
  • $$620$$
Find the sum of the following series to n terms:
$$1\times 2+2\times 3+3\times 4+4\times 5+...$$
  • $$\dfrac{n}{4}(n-1)(n+2)$$
  • $$\dfrac{n}{3}(n-1)(n-2)$$
  • $$\dfrac{n}{2}(n-1)(n+1)$$
  • $$\dfrac{n}{3}(n+1)(n+2)$$
Find the sum of the following series to n terms:
$$1+(1+2)+(1+2+3)+(1+2+3+4)+...$$
  • $$\dfrac{n}{6}(n+1)(n+2)$$
  • $$\dfrac{n}{6}(n-1)(n-2)$$
  • $$\dfrac{n}{6}(n+1)(n-1)$$
  • None of these
5,5,5,... forms a ______
  • Series
  • Sequence
  • Progression
  • None of the above
Whole numbers from a sequence/progression, as they are formed by the fixed rule of adding 1 to previous whole number.
  • True
  • False
State true/false:
$$3,4,5,6,7,......$$ forms a progression. As they are formed by the rule $$n+1$$
  • True
  • False
Find the sum of Arithmetic means of $$3 , 9$$ and $$12 , 8$$.
  • $$6$$
  • $$9$$
  • $$20$$
  • $$16$$
The arithmetic mean of $$4$$ and $$14$$ is 
  • $$9$$
  • $$18$$
  • $$10$$
  • $$4$$
If AM of two numbers a and 17 isThen a is ___
  • 15
  • 13
  • 17
  • None of the above
Insert an arithmetic mean between $$7$$ and $$21$$
  • $$10$$
  • $$14$$
  • $$28$$
  • $$30$$
Arithmetic mean of two numbers a+d and a-d is 
  • $$a$$
  • $$b$$
  • Cannot be determined
  • None of the above
Sum of the series $$S=1^2-2^2+3^2-4^2+.....-2002^2+2003^2$$ is
  • $$2007006$$
  • $$1005004$$
  • $$2000506$$
  • $$1005040$$
The sum of $$n$$ terms of the series $$1^2-2^2+3^2-4^2+5^2-6^2+...$$ is 
  • $$\dfrac{-n(n+1)}{2}$$ if $$n$$ is even
  • $$\dfrac{n(n+1)}{2}$$ if $$n$$ is odd
  • $$-n(n+1)$$ if $$n$$ is even
  • $$\dfrac{n(n+1)(2n+1)}{6}$$ if $$n$$ is odd
$$\displaystyle \frac{1^{3}}{1}+\frac{1^{3}+2^{3}}{1+3}+\frac{1^{3}+2^{3}+3^{3}}{1+3+5}+\ldots.n$$ terms $$=$$
  • $$\displaystyle \frac{\mathrm{n}(2\mathrm{n}^{2}+9\mathrm{n}+13)}{24}$$
  • $$\displaystyle \frac{n(2n^{3}+9n+13)}{8}$$
  • $$\displaystyle \frac{n(n^{2}+9n+13)}{24}$$
  • $$\displaystyle \frac{n(n^{2}+9n+13)}{8}$$

Observe the following lists List I and  List II

(A) $$\displaystyle \sum_{n=0}^{\infty}\frac{x^{n}(\log_{e}a)^{n}}{n!}$$                                     $$(1)\displaystyle \frac{e^{x}-e^{-x}}{2}$$
(B) $$\displaystyle \sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}$$                                                $$(2) e^{-ax}$$
(C) $$\displaystyle \sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}=$$                                    $$(3) a^{x}$$
(D) $$\displaystyle \sum_{n=0}^{\infty}\frac{(-1)^{n}.(ax)^{n}}{n!}$$                                  $$(4)\displaystyle \frac{a^{x}-a^{-x}}{2}$$
                                                                       $$(5)\displaystyle \frac{e^{x}+e^{-x}}{2}$$
The correct match of List I to List II is:
  • A - 1, B - 4, C - 3, D - 5
  • A - 4, B - 2, C - 1, D - 3
  • A - 3, B - 5, C - 1, D - 2
  • A - 2, B - 3, C - 5, D - 4
 Sum  the  series  $$1^3+3^3+5^3+.......... $$  to  $$n$$ terms  is
  • $$ n^2\left ( 2n^2-1 \right )$$
  • $$n\left ( 2n^2-1 \right )$$
  • $$n\left ( 2n^2+1 \right )$$
  • $$n^2\left ( 2n^2+1 \right )$$
If $$ \displaystyle f(r)=1+\frac {1}{2}+\frac {1}{3}+.....+\frac {1}{r}$$ and $$f(0)=0$$, then value of $$ \displaystyle \sum_{r=1}^{n}(2r+1)f(r)$$ is
  • $$n^2f(n)$$
  • $$ \displaystyle (n+1)^2f(n+1)-\frac {n^2+3n+2}{2}$$
  • $$ \displaystyle (n+1)^2f(n)-\frac {n^2+n+1}{2}$$
  • $$(n+1)^2f(n)$$
If $$n$$ is an odd integer greater than or equal to $$1$$, then the value of $$ n^3-(n-1)^3+(n-2)^3-....+(-1)^{n-1}1^3$$, is
  • $$\dfrac{(n+1)^2(2n-1)}{4}$$
  • $$ \dfrac{(n-1)^2(2n-1)}{4}$$
  • $$\dfrac{(n+1)^2(2n+1)}{4}$$
  • $$\dfrac{(n+1)^2(2n+1)}{8}$$
The sum to $$n$$ terms of the series $$\displaystyle \frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+...$$ is
  • $$\displaystyle \frac{6n}{n+1}$$
  • $$\displaystyle \frac{9n}{n+1}$$
  • $$\displaystyle \frac{12n}{n+1}$$
  • $$\displaystyle \frac{3n}{n+1}$$
The sum of the first $$n$$ terms of the series $$1^2+2\cdot2^2+3^2+2\cdot4^2+5^2+2\cdot6^2+..... $$ is $$\dfrac{n \left ( n+1\right )^2}{2}$$ when $$n$$ is even. 
When $$n$$ is odd, then the sum is
  • $$ \dfrac{3n \left ( n+1\right )}{2}$$
  • $$ \dfrac{n (n+1)^2}{4}$$
  • $$\dfrac{n \left ( n+1\right )^2}{2}$$
  • $$\dfrac{n^2 \left ( n+1\right )}{2}$$
The sum of n terms of the series $$1 + (1 + a) + (1 + a + a^2) + (1 + a  + a^2 + a^3) + ....$$ is
  • $$\displaystyle \frac{n}{1+a} - \frac{1 - a^n}{(1 - a)^2}$$
  • $$\displaystyle \frac{n}{1- a} + \frac{a (1 - a^n)}{(1 - a)^2}$$
  • $$\displaystyle \frac{n}{1 - a} + \frac{a (1 + a^n)}{(1 - a)^2}$$
  • none of the above
ABCD is a square of length a, $$a\epsilon N$$, a>Let $$L_{1}$$, $$L_{2}$$, $$L_{3}$$,... be points on BC such that $$BL_{1}=L_{1}L_{2}=L_{2}L_{3}=...=1$$ and $$M_{1}$$, $$M_{2}$$, $$M_{3}$$,... be points on CD such that $$CM_{1}=M_{1}M_{2}=M_{2}M_{3}=...=1$$. Then $$\sum_{n=1}^{a-1}\left ( A{L_{n}}^{2}+L_{n}{M_{n}}^{2} \right )$$ is equal to
  • $$\dfrac{1}{2}a\left ( a-1 \right )^{2}$$
  • $$\dfrac{1}{2}a\left ( a-1 \right )\left ( 4a-1 \right )$$
  • $$\dfrac{1}{2}a\left ( a-1 \right )\left ( 2a-1 \right )\left ( 4a-1 \right )$$
  • none of these
If $$\displaystyle \sum_{r=1}^{n}t_{r}=\frac {n(n+1)(n+2)(n+3)}{8}$$, then $$\displaystyle \underset{n\rightarrow \infty}{ \lim} \sum_{r=1}^{n}\frac {1}{t_{r}}$$
is equal to: 
  • $$\displaystyle \frac {1}{8}$$
  • $$\displaystyle \frac {1}{4}$$
  • $$\displaystyle \frac {1}{2}$$
  • $$1$$
Let x, y, z be three positive prime numbers. The progression in which $$\sqrt{x}$$, $$\sqrt{y}$$, $$\sqrt{z}$$ can be three terms (not necessarily consecutive) is
  • AP
  • GP
  • HP
  • none of these
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