CBSE Questions for Class 11 Engineering Maths Sequences And Series Quiz 6 - MCQExams.com

$$6, 10, 18, 34, 66$$
The first number in the list above is $$6$$. Determine a rule for finding each successive number in the list.
  • Add $$4$$ to the preceding number.
  • Take $$\displaystyle \frac { 1 }{ 2 } $$ of the preceding number and then add $$7$$ to that result.
  • Double the preceding number and then subtract $$2$$ from that result.
  • Subtract $$2$$ from the preceding number and then double that result.
  • Triple the preceding number and then subtract $$8$$ from that result.
The value of the sum $$1.2.3+2.3.4+3.4.5+...$$ upto n terms =
  • $$\dfrac{1}{6}n^2 (2n^2+1)$$
  • $$\dfrac{1}{6}(n^2-1)(2n-1)(2n+3)$$
  • $$\dfrac{1}{8}(n^2+1)(n^2+5)$$
  • $$\dfrac{1}{4}n(n+1)(n+2)(n+3)$$
$$m, 2m, 4m, . . . $$
The first term in the sequence above is $$m$$, and each term thereafter is equal to twice the previous term. If $$m$$ is an integer, which of the following could NOT be the sum of the first four terms of this sequence?
  • $$-26$$
  • $$-15$$
  • $$45$$
  • $$75$$
  • $$120$$
Identify the missing integer: $$9, 45,$$ ____$$, 1125, 5625...$$
  • $$220$$
  • $$223$$
  • $$224$$
  • $$225$$
The terms of a sequence are defined by $$a_{n} = 3a_{n - 1} - a_{n - 2}$$ for $$n > 2$$. Find the value of $$a_{5}$$ given that $$a_{1} =4$$ and $$a_{2} = 3$$.
  • $$12$$
  • $$23$$
  • $$25$$
  • $$31$$
  • $$36$$
For all numbers a and b, let $$\displaystyle a\bigodot b$$ be defined by $$\displaystyle a\bigodot b=ab+a+b$$. Then for the numbers $$x$$, $$y$$ and $$z$$, which of the following is/are true?
I. $$\displaystyle x\bigodot y=y\bigodot x$$
II. $$\displaystyle \left( x-1 \right) \bigodot \left( x+1 \right) =\left( x\bigodot x \right) -1$$
III. $$\displaystyle x\bigodot \left( y+z \right) =\left( x\bigodot y \right) +\left( x\bigodot z \right) $$
  • I only
  • II only
  • III only
  • I and II only
  • I, II, and III
The sum of $$1$$ st $$n$$ terms of the series
$$\cfrac { { 1 }^{ 2 } }{ 1 } +\cfrac { { 1 }^{ 2 }+{ 2 }^{ 2 } }{ 1+2 } +\cfrac { { 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 } }{ 1+2+3 } +........$$
  • $$\cfrac { n+2 }{ 3 } $$
  • $$\cfrac {n( n+2 )}{ 3 } $$
  • $$\cfrac {n( n-2 )}{ 3 } $$
  • $$\cfrac {n( n-2 )}{ 6 } $$
If $$m * n = m+(m-1)+(m-2)+ ...... +(m-n)$$, evaluate $$7 * 5$$.
  • $$25$$
  • $$20$$
  • $$27$$
  • $$18$$
$$N=a^2 + b^2$$ is a three-digit number which is divisible bya = 10x + y and b = 10x + z, where z is a prime number, and x and y are natural numbers. If a + b = 31, find the value of N.
  • 565
  • 485
  • 505
  • 485 or 505
If $$a\odot b = 6\times a - 3\times b$$, evaluate $$(5\odot 3) \odot 20$$
  • $$2$$
  • $$41$$
  • $$66$$
  • $$1$$
The sum of the series,
$$\displaystyle \frac{1}{2.3}\cdot 2+ \frac{2}{3.4}\cdot 2^2+\frac{3}{4.5}\cdot 2^3+ ......$$ to n terms is _____.
  • $$\dfrac{2^{n+1}}{n+2}+1$$
  • $$\dfrac{2^{n+1}}{n+2}-1$$
  • $$\dfrac{2^{n+1}}{n+2}+2$$
  • $$\dfrac{2^{n+1}}{n+2}-2$$
The value of the infinite series $$\cfrac{1^2+2^2}{\underline{|3}} + \cfrac{1^2+2^2+3^2}{\underline{|4}}+\cfrac{1^2+2^2+3^3+4^2}{\underline{|5}}........$$ is
  • $$e$$
  • $$5e$$
  • $$\cfrac{5e}{6} -\cfrac{1}{2}$$
  • $$\cfrac{5e}{6}$$
Let $$S$$ denote the sum of the infinite series $$1+\cfrac { 8 }{ 2! } +\cfrac { 21 }{ 3! } +\cfrac { 40 }{ 4! } +\cfrac { 65 }{ 5! } +.......$$. Then
  • $$S< 8$$
  • $$S> 12$$
  • $$8< S< 12$$
  • $$S=8$$
$$x=1+\frac{1}{2\times \underline{|1}}+\frac{1}{4\times \underline{|2}}+\frac{1}{8\times \underline{|3}}$$
  • $${e}^{1/2}$$
  • $$e^2$$
  • e
  • $$\frac{1}{e}$$
Find the $$(2n)^{th}$$ term of the series whose $$n^{th}$$ term is $$\dfrac{n^2+1}{n^3}$$:
  • $$\dfrac{n^2+1}{8n^3}$$
  • $$\dfrac{4n^2+1}{8n^3}$$
  • $$\dfrac{4n^2+1}{n^3}$$
  • $$\dfrac{2n^2+1}{2n^3}$$
In the series $$1+3+6+10+......,$$ find the $$n^{th}$$ term:
  • $$\dfrac{n(n-1)}{2}$$
  • $$\dfrac{n(n+1)}{2}$$
  • $$\dfrac{n(2n-1)}{2}$$
  • $$\dfrac{n(2n+1)}{2}$$
Consider an incomplete pyramid of balls on a square base having $$18$$ layers; and having $$13$$ balls on each side of the top layer. Then the total number $$N$$ of balls in that pyramid satisfies
  • $$9000 < N < 10000$$
  • $$8000 < N < 9000$$
  • $$7000 < N < 8000$$
  • $$10000 < N < 12000$$
With the help of match-sticks, Zalak prepared a pattern as shown below. When $$97$$ matchsticks are used, the serial number of the figure will be ...........
626028_e327aecd938b484db21d45ffb46b1a27.PNG
  • Figure 32
  • Figure 95
  • Figure 49
  • Figure 48
Three bells commenced to toll at the same time and tolled at intervals of $$20, 30, 40$$ seconds respectively. If they toll together at $$6$$ am, then which of the following is the time at which they can toll together
  • $$6:55$$am
  • $$6:56$$am
  • $$6:57$$am
  • $$6:59$$am
Let $$a_n=\dfrac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}$$ then $$\displaystyle \sum_{n=1}^{144}a_n$$ equals
  • $$2456$$
  • $$2645$$
  • $$2466$$
  • $$2546$$
If $$\displaystyle \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + ..... $$ upto $$\displaystyle \infty = \frac{\pi^2}{6},$$ then $$\displaystyle \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + .... =$$
  • $$\dfrac{\pi^2}{12}$$
  • $$\dfrac{\pi^2}{24}$$
  • $$\dfrac{\pi^2}{8}$$
  • $$\dfrac{\pi^2}{4}$$
$$\displaystyle \sum_{k = 1}^{\infty} \dfrac {6^{k}}{(3^{2k + 1} + 2^{2k + 1}) - (3^{k}2^{k + 1} + 2^{k}3^{k + 1})}$$ is equal to
  • $$3$$
  • $$\dfrac {1}{3}$$
  • $$\dfrac {4}{3}$$
  • $$\dfrac {6}{5}$$
Let $$S = \displaystyle \sum_{n = 1}^{99} = \dfrac {5^{100}}{5^{100} + 25^{n}}$$ then find the value of $$[S]$$, where $$[.] = G.I.F.$$
  • $$99$$
  • $$100$$
  • $$25$$
  • $$49$$
The first term of an AP is $$148$$ and the common difference is $$-2$$. If the AM of first $$n$$ terms of the AP is $$125$$, then the value of $$n$$ is
  • $$18$$
  • $$24$$
  • $$30$$
  • $$36$$
  • $$48$$
The sum of the series $$\displaystyle \sum_{n = 8}^{17} \dfrac{1}{(n + 2)(n + 3)} $$ is equal to
  • $$\dfrac{1}{17}$$
  • $$\dfrac{1}{18}$$
  • $$\dfrac{1}{19}$$
  • $$\dfrac{1}{20}$$
  • $$\dfrac{1}{21}$$
The value of $$\dfrac {1}{i} + \dfrac {1}{i^{2}} + \dfrac {1}{i^{3}} + .... + \dfrac {1}{i^{102}}$$ is
  • $$-1 - i$$
  • $$-1 + i$$
  • $$1 - i$$
  • $$1 + i$$
  • $$1 - 2i$$
The sum of the first $$n$$ terms of the series $${ 1 }^{ 2 }+2\cdot { 2 }^{ 2 }+{ 3 }^{ 3 }+2\cdot { 4 }^{ 2 }+{ 5 }^{ 2 }+2\cdot { 6 }^{ 2 }+\cdots $$ is $$\dfrac { n{ \left( n+1 \right)  }^{ 2 } }{ 2 } $$ when $$n$$ is even, when $$n$$ is odd the sum is
  • $$\dfrac { 3n\left( n+1 \right) }{ 2 } $$
  • $$\dfrac { { n }^{ 2 }\left( n+1 \right) }{ 2 } $$
  • $$\dfrac { n{ \left( n+1 \right) }^{ 2 } }{ 4 } $$
  • $${ \left[ \dfrac { n\left( n+1 \right) }{ 2 } \right] }^{ 2 }$$
Find the sum of the series 
$$\displaystyle \frac{1}{2\cdot 3}+\frac {1}{4\cdot 5}+\frac {1}{6\cdot 7}+ ...$$
  • $$log\, \frac {e}{2}$$
  • $$log\, \frac {e}{4}$$
  • $$log\, \frac {2}{3}$$
  • $$log\, \frac {2}{4}$$
If the natural numbers are divided into groups of {1}, {2, 3}, {4, 5, 6}, {7, 8, 9, 10} ....Then the /sum of 50th group is 
693570_f1a93ed930d240ae89ce071c94c93b40.PNG
  • 65225
  • 56225
  • 62525
  • 53625
The value of a for which side of nth square equals the diagonals of $$(n + 1)^{th}$$ square is 
  • 1/3
  • 1/4
  • 1/2
  • $$1\sqrt{2}$$
If $$\displaystyle \sum_{r = 1}^{n}t_{n} = \dfrac {n(n +1)(n + 2)(n + 3)}{8}$$, then $$\displaystyle \sum_{r = 1}^{n} \dfrac {1}{t_{1}}$$ equals
  • $$-\left (\dfrac {1}{(n + 1)(n + 2)} - \dfrac {1}{2}\right )$$
  • $$\left (\dfrac {1}{(n + 1)(n + 2)} - \dfrac {1}{2}\right )$$
  • $$\left (\dfrac {1}{(n + 1)(n + 2)} + \dfrac {1}{2}\right )$$
  • $$\left (\dfrac {1}{(n - 1)(n - 2)} + \dfrac {1}{2}\right )$$
Let $$r^{th} $$ term of a series is given by, $$T_r = \dfrac {r}{1-3r^2 + r^4} .$$
Then $$ \underset {n \rightarrow \infty}{\lim} \sum_{r=1}^n T_r $$ is 
  • $$\cfrac { 3 }{ 2 } $$
  • $$\cfrac { 1 }{ 2 } $$
  • $$\cfrac { -1 }{ 2 } $$
  • $$\cfrac { -3 }{ 2 } $$
If $$\alpha =1 / 4$$ and $$P_n$$ denotes the perimeter of the nth square then$$\sum_{n=1}^{\infty } P_n$$ equals 
  • $$\dfrac{8}{3}\left ( 4+\sqrt{10} \right )$$
  • $$\dfrac{8}{3}$$
  • $$\dfrac{16}{3}$$
  • None of these
If $$ a_1 \in R - \left \{ 0 \right \}, i = 1, 2, 3, 4$$ and $$x \in R$$ and $$\left ( \sum_{i=1}^{3}a_i^2\right ) x^2-2x \left ( \sum_{i=1}^{3}a_i a_{i+1}\right )+\sum_{i=2}^{4}a_i^2\leq 0$$, then $$a_1, a_2, a_3, a_4$$ are in
  • A.P
  • G.P
  • H.P
  • A.G.P
If $${ b }_{ i }=1-{ a }_{ i },na=\sum _{ i=1 }^{ n }{ { a }_{ i } } ,nb=\sum _{ i=1 }^{ n }{ { b }_{ i } } \quad $$, then $$\sum _{ i=1 }^{ n }{ { { a }_{ i }b }_{ i } } +\sum _{ i=1 }^{ n }{ { \left( { a }_{ i }-a \right)  }^{ 2 } } =$$
  • $$ab$$
  • $$-nab$$
  • $$nab$$
  • $$(n+1)ab$$
$$4,9,25,?,121,169$$
  • $$36$$
  • $$49$$
  • $$64$$
  • $$81$$
Let S be the infinite sum given by $$S=\displaystyle \sum_{n=0}^{\infty}\frac{a_n}{10^{2n}}$$, where $$(a_n)_{n\geq 0}$$ is a sequence defined by $$a_0=a_1=1$$ and $$a_j=20a_{j-1}$$ for $$j\geq 2$$. If $$S$$ is expressed in the form $$\displaystyle\frac{a}{b}$$, where $$a, b$$ are coprime positive integers, than $$a$$ equals.
  • $$60$$
  • $$75$$
  • $$80$$
  • $$81$$
Find the missing number in the circle:
716190_432c83cc262449fb9c49be0336e200ad.jpg
  • $$66$$
  • $$72$$
  • $$71$$
  • $$78$$
$$\displaystyle \sum_{r = 0}^{n}{ \left( \frac{ 2^{r-2} . ^nC_r}{(r+1)(r+2)} \right) }$$ is equal to
  • $$\displaystyle \frac{3^{n+2} - 2n + 5}{(n+1)(n+2)}$$
  • $$\displaystyle \frac{3^{n+2} - 4n + 5}{(n+1)(n+2)}$$
  • $$\displaystyle \frac{3^{n+2} - 2n - 5}{(n+1)(n+2)}$$
  • None of these
13, 74, 290, 650,.......
  • $$1248$$
  • $$1370$$
  • $$1346$$
  • $$1452$$
  • $$1625$$
Figures $$1$$ and $$2$$ are related in a particular manner. Establish the same relationship between figures $$3$$ and $$4$$ by choosing a figure from amongst the options.
726593_93a844c1ede949b2b2fa594a3d0af569.png
Select the INCORRECT match
  • $$3249-MMMCCXLIX$$
  • $$1667-MDCLXVII$$
  • $$207-CCXVII$$
  • $$499-CDXCIX$$
The value of $$\sum _{ n=1 }^{ \infty  }{ { \left( -1 \right)  }^{ n+1 }\left( \cfrac { n }{ { 5 }^{ n } }  \right)  } $$ equals
  • $$\cfrac { 5 }{ 12 } $$
  • $$\cfrac { 5 }{ 24 } $$
  • $$\cfrac { 5 }{ 36 } $$
  • $$\cfrac { 5 }{ 16 } $$
If the $$p^{th}$$ term of the series of positive numbers $$25, 22\dfrac {3}{5}, 20\dfrac {1}{2}, 18\dfrac {1}{4}$$, .... is numerically the smallest, then the $$p^{th}$$ is.
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{7}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{5}$$
The sum of first $$20$$ terms of the series $$1,6,13,22$$- is
  • $$5580$$
  • $$5780$$
  • $$7789$$
  • $$1237$$
If $$\begin{vmatrix} x \end{vmatrix}<1$$  then the coefficient of $$x^5$$ in the expansion of $$\dfrac{3x}{(x-2) (x-1)}$$ is
  • $$\dfrac{33}{32}$$
  • $$-\dfrac{33}{32}$$
  • $$\dfrac{31}{32}$$
  • $$-\dfrac{33}{34}$$
$$2.4+4.7+6.10+.....$$ upto $$(n-1)$$ terms 
  • $$2{ n }^{ 3 }+2{ n }^{ 2 }$$
  • $$\cfrac { 1 }{ 6 } \left( { n }^{ 3 }+3{ n }^{ 2 }+1 \right) $$
  • $$2{ n }^{ 3 }-2{ n }^{ 2 }$$
  • $$\cfrac { 1 }{ 6 } \left( { n }^{ 2 }-3{ n }+1 \right) $$
Sum of the series $$\sum _{ r=1 }^{ n }{ \left( { r }^{ 2 }+1 \right) r! } $$ is ______
  • $$(n+1)!$$
  • $$(n+2)!-1$$
  • $$n(n+1)!$$
  • none of these
Given $$S_n=1+q+q^2+...+q^n$$ & $$S_n=1+\displaystyle\frac{q+1}{2}+\left(\frac{q+1}{2}\right)^2+....+\left(\displaystyle\frac{q+1}{2}\right)^n, q\neq 1$$. then $$^{n+1}C_1+ {^{n+1}C_2\cdot s_1}+ {^{n+1}C_3}\cdot s_2+....+{^{n+1}C_{n+1}}\cdot s_n=2^n\cdot S_n$$.
  • True
  • False
Suppose $$a_2,a_3,a_4,a_5,a_6,a_7$$ are integers such that
$$\dfrac {5}{7}=\dfrac {a_2}{2!}+\dfrac {a_3}{3!}+\dfrac {a_4}{4}+\dfrac {a_5}{5!}+\dfrac {a_6}{6!}+\dfrac{a_7}{7!}$$
where $$0 \le a < j$$ for $$ j=2,4,5,6,7.$$ The sum $$a_2+a_3+a_4+a_5+a_6+a_7$$ is 
  • 8
  • 9
  • 10
  • 11
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