Sequences and Series jee mains questions | Sequences and Series Previous Year Question

Q 21 :

The interior angles of a polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number of sides of the polygon is

9
10
16
5

1

2

3

4

(1)

$Let angles of a polygon are a, a + d, a + 2 d, \dots$

$a = 120 °,$ $d = 5$

$So series are 120 °, 125 °, 130 °, \dots$

$Sum of A.P. = (n - 2) 180 °$

$\Rightarrow$ $\frac{n}{2} [2 a + (n - 1) d] = (n - 2) 180 °$

$\Rightarrow$ $\frac{n}{2} [240 + (n - 1) 5] = 180 (n - 2)$

$\Rightarrow$ $5 n^{2} - 125 n + 720 = 0 \Rightarrow n^{2} - 25 n + 144 = 0$

$\Rightarrow$ $n = 16,$ $n = 9 not possible.$

Q 22 :

The sum to infinity of the series $1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \frac{14}{3^{4}} + \dots$ is

3
4
6
2

1

2

3

4

(1)

$Let S = 1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \frac{14}{3^{4}} + \dots$ ...(i)

$\frac{1}{3} S = \frac{1}{3} + \frac{2}{3^{2}} + \frac{6}{3^{3}} + \frac{10}{3^{4}} + \dots$ ...(ii)

$Subtracting (ii) from (i), we get$

$\frac{2}{3} S = 1 + \frac{1}{3} + \frac{4}{3^{2}} + \frac{4}{3^{3}} + \frac{4}{3^{4}} + \dots$

$= \frac{4}{3} [1 + \frac{1}{3} + \dots to \infty] = \frac{4}{3} \cdot \frac{1}{1 - \frac{1}{3}} = \frac{4}{3} \cdot \frac{1}{\frac{2}{3}} = 2$

$\Rightarrow$ $\frac{2}{3} S = 2 \Rightarrow S = 3$

Q 23 :

In a G.P., $t_{2} + t_{5} = 216$ and $t_{4} : t_{6} = 1 : 4$ and all terms are integers, then its first term is

16
14
12
none of these

1

2

3

4

(3)

$Let G.P. is a, a r, a r^{2}, a r^{3}, a r^{4}, \dots$

$∴$ $t_{2} + t_{5} = a r + a r^{4} = 216$

$and$ $\frac{t_{4}}{t_{6}} = \frac{a r^{3}}{a r^{5}} = \frac{1}{4} \Rightarrow r^{2} = 4 \Rightarrow r = \pm 2$

$For r = 2$

$For r = - 2$

$\begin{matrix} a (2 + 2^{4}) = 216 \\ \Rightarrow a (18) = 216 \\ \Rightarrow a = \frac{216}{18} = 12 \end{matrix} | \begin{matrix} a (- 2 + 2^{4}) = 216 \\ \Rightarrow a (14) = 216 \\ \Rightarrow a = \frac{216}{14} = \frac{108}{7} \end{matrix}$

$∴$ $a = 12$ $(since all terms are integers)$

Q 24 :

If $3^{rd}$ , $7^{th}$ and $12^{th}$ terms of an A.P. are three consecutive terms of a G.P., then the common ratio of the G.P. is

$\frac{5}{4}$
$\frac{9}{4}$
$\frac{2}{9}$
$\frac{1}{2}$

1

2

3

4

(1)

Q 25 :

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

$3 + \sqrt{2}$
$2 - \sqrt{3}$
$2 + \sqrt{3}$
$\sqrt{2} + \sqrt{3}$

1

2

3

4

(3)

$Let the numbers be a, a r, a r^{2}, we have$

$2$ $| 2 a r |$ $= a + a r^{2} \Rightarrow 4 r = r^{2} + 1 \Rightarrow r^{2} - 4 r + 1 = 0$

$\Rightarrow r = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2 \sqrt{3}}{2} = 2 \pm \sqrt{3}$

$∴$ $r = 2 + \sqrt{3}, as G.P. is increasing.$

Q 26 :

If ${(10)}^{9} + 2 {(11)}^{1} {(10)}^{8} + 3 {(11)}^{2} {(10)}^{7} + \dots + 10 {(11)}^{9} = k {(10)}^{9}$ , then $k$ is equal to

$\frac{441}{100}$
$100$
$110$
$\frac{121}{100}$

1

2

3

4

(2)

$Let P = 10^{9} + 2 \cdot 11 \cdot 10^{8} + \dots + 10 \cdot 11^{9}$

$We have, \frac{11}{10} P = 11 \cdot 10^{8} + \dots + 9 \cdot 11^{9} + 11^{10}$

$On subtracting, we get$

$\Rightarrow \frac{1}{10} P = 11^{10} - [10^{9} + 11^{1} \cdot 10^{8} + 11^{2} \cdot 10^{7} + \dots + 11^{9}]$

$= 11^{10} - 10^{9} {\frac{{(\frac{11}{10})}^{10} - 1}{\frac{11}{10} - 1}} = 11^{10} - 11^{10} + 10^{10}$

$∴$ $P = 10^{11} = (100) \cdot 10^{9}$

$On comparison, k = 100$

Q 27 :

If $a, b, c$ are in H.P., then $\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}$ will be in

A.P.
G.P.
H.P.
None of these

1

2

3

4

(3)

$Since, a, b, c are in H.P.$

$\Rightarrow$ $\frac{1}{a}, \frac{1}{b}, \frac{1}{c} are in A.P.$

$\Rightarrow$ $\frac{a + b + c}{a}, \frac{a + b + c}{b}, \frac{a + b + c}{c} are in A.P.$

$\Rightarrow$ $1 + \frac{b + c}{a}, 1 + \frac{a + c}{b}, 1 + \frac{a + b}{c} are in A.P.$

$\Rightarrow$ $\frac{b + c}{a}, \frac{a + c}{b}, \frac{a + b}{c} are in A.P.$

$\Rightarrow$ $\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b} are in H.P.$

Q 28 :

If A, G and H are respectively the A.M., the G.M. and the H.M. between two positive numbers $a$ and $b$ , then the correct relation is

$A = G^{2} H$
$A = G H^{2}$
$A^{2} = G H$
$G^{2} = A H$

1

2

3

4

(4)

$For positive numbers a and b,$

$A = \frac{a + b}{2},$ $G = \sqrt{a b}$ $and$ $H = \frac{2 a b}{a + b}$

$Now,$ $A H = \frac{a + b}{2} (\frac{2 a b}{a + b}) = a b = G^{2}$

$∴$ $The correct relation is, G^{2} = A H$

Q 29 :

The positive numbers $a_{1}, a_{2}, a_{3}$ are in A.P. such that $a_{1} a_{2} a_{3} = 64$ . Then the minimum value of $a_{2}$ is

4
8
16
32

1

2

3

4

(1)

$From A.M.-G.M. theorem$

$\frac{a_{1} + a_{2} + a_{3}}{3} \geq \sqrt[3]{a_{1} a_{2} a_{3}}$

$\Rightarrow$ $a_{1} + a_{2} + a_{3} \geq 3 \sqrt[3]{64}$

$\Rightarrow$ $a_{1} + a_{2} + a_{3} \geq 12$ ...(i)

$Now, a_{2} is the mean of a_{1} and a_{3}$

$∴$ $a_{1} + a_{3} = 2 a_{2}$ ...(ii)

$So from (i) and (ii)$

$2 a_{2} + a_{2} \geq 12 \Rightarrow 3 a_{2} \geq 12 \Rightarrow a_{2} \geq 4$

$∴$ ${(a_{2})}_{\min} = 4$

Q 30 :

If A.M. and G.M. of $x$ and $y$ are in the ratio $p : q$ , then $x : y$ is

$p - \sqrt{p^{2} + q^{2}} : p + \sqrt{p^{2} + q^{2}}$
$p + \sqrt{p^{2} - q^{2}} : p - \sqrt{p^{2} - q^{2}}$
$p : q$
$p + \sqrt{p^{2} + q^{2}} : p - \sqrt{p^{2} + q^{2}}$

1

2

3

4

(2)

$A = \frac{x + y}{2}$ $and$ $G = \sqrt{x y}$

$x + y = 2 A$ $and$ $G^{2} = x y$

$The equation having x, y as its roots is$

$t^{2} - (x + y) t + x y = 0$ $or,$ $t^{2} - 2 A t + G^{2} = 0$

$∴$ $t = \frac{2 A \pm \sqrt{4 A^{2} - 4 G^{2}}}{2} \Rightarrow t = A \pm \sqrt{A^{2} - G^{2}}$

$So, the two numbers are$

$x = A + \sqrt{A^{2} - G^{2}}$ $and$ $y = A - \sqrt{A^{2} - G^{2}}$

$It is given that A : G = p : q \Rightarrow A = λ p$ $and$ $G = λ q$

$Now, substituting the values of A and G in$

$x = A + \sqrt{A^{2} - G^{2}}$ $and$ $y = A - \sqrt{A^{2} - G^{2}}, we get$

$\frac{x}{y} = \frac{λ p + \sqrt{λ^{2} p^{2} - λ^{2} q^{2}}}{λ p - \sqrt{λ^{2} p^{2} - λ^{2} q^{2}}} \Rightarrow \frac{x}{y} = \frac{p + \sqrt{p^{2} - q^{2}}}{p - \sqrt{p^{2} - q^{2}}}$

$∴$ $x : y = (p + \sqrt{p^{2} - q^{2}}) : (p - \sqrt{p^{2} - q^{2}})$

Q 31 :

The sum of the first 8 terms of $\frac{1^{2}}{2} + \frac{1^{2} + 2^{2}}{3} + \frac{1^{2} + 2^{2} + 3^{2}}{4} + \dots$ is

72
73
74
75

1

2

3

4

(3)

Q 32 :

If the $10^{th}$ term of an A.P. is $\frac{1}{20}$ and its $20^{th}$ term is $\frac{1}{10}$ , then the sum of its first 200 terms is

$50 \frac{1}{4}$
$100$
$50$
$100 \frac{1}{2}$

1

2

3

4

(4)

$Given, T_{10} = \frac{1}{20} = a + 9 d$ ...(i)

$and$ $T_{20} = \frac{1}{10} = a + 19 d$ ...(ii)

$Solving (i) and (ii), we have$

$a = \frac{1}{200},$ $d = \frac{1}{200}$

$∴$ $S_{200} = \frac{200}{2} [\frac{2}{200} + \frac{199}{200}] = \frac{201}{2} = 100 \frac{1}{2}$

Q 33 :

If the $19^{th}$ term of a non-zero A.P. is zero, then its ( $49^{th}$ term) : ( $29^{th}$ term) is

3 : 1
4 : 1
2 : 1
1 : 3

1

2

3

4

(1)

$Given, 19^{th} term of a non-zero A.P. = 0$

$∴$ $a + 18 d = 0,$ where $a$ be the first term and $d$ be common difference of A.P.

$Now, \frac{49^{th} term}{29^{th} term} = \frac{a + 48 d}{a + 28 d} = \frac{a + 18 d + 30 d}{a + 18 d + 10 d} = \frac{3}{1}$

Q 34 :

The sum of all those terms, of the arithmetic progression 3, 8, 13, ....., 373, which are not divisible by 3, is equal to ______.

(9525)

$The given A.P. is 3, 8, 13, \dots, 373$

$T_{n} = a + (n - 1) d \Rightarrow 373 = 3 + (n - 1) 5 \Rightarrow n = 75$

$S_{n} = \frac{75}{2} [3 + 373] = 14100$

$Numbers which are divisible by 3 are 3, 18, \dots, 363$

$T_{n'} = 3 + (n' - 1) 15$

$\Rightarrow 363 = 3 + 15 n' - 15 \Rightarrow 15 n' = 375 \Rightarrow n' = 25$

$S_{n'} = \frac{25}{2} [3 + 363] = 4575$

$∴$ $Required sum = 14100 - 4575 = 9525$

Q 35 :

Let the digits $a, b, c$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

(1260)

$There are only two ways to write digits a, b, c, be in A.P.$

$\begin{matrix} I & a b c \\ I I & c b a \end{matrix}$ $\begin{matrix} = & {}^{2}C_{1} \end{matrix}$

---------------------------------------

Here, 9 places to put $a b c$ or $b c a$ but there will be only 7 possible ways A.P. to choose three consecutive numbers are

$i.e., 123, 234, 345, 456, 567, 678, 789 = {}^{7}C_{1}$

Now, we have left only 6 places where $a, b, c$ three such that three consecutive digits are in A.P.

$\Rightarrow \frac{6!}{2! 2! 2!}$

$Required number = {}^{2}C_{1} \cdot {}^{7}C_{1} \cdot \frac{6!}{2! 2! 2!}$

$= \frac{2 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2}{2 \times 2 \times 2} = 1260$

Q 36 :

The sum of the common terms of the following three arithmetic progressions 3, 7, 11, 15, ...., 399; 2, 5, 8, 11, ...., 359 and 2, 7, 12, 17, ...., 197, is equal to ______.

(321)

$3, 7, 11, 15, \dots, 399 are in A.P.$

So, common difference $d_{1} = 7 - 3 = 4$

$2, 5, 8, 11, \dots, 359 are in A.P.$

Then, $d_{2} = 5 - 2 = 3; 2, 7, 12, 17, \dots, 197 are in A.P.$

$d_{3} = 7 - 2 = 5$

$L . C . M . of d_{1}, d_{2}, d_{3} = L . C . M . of (4, 3, 5) = 60$

On expanding all A.P.s, common terms are $47, 107, 167$

$∴$ $Sum = 47 + 107 + 167 = 321$

Q 37 :

If $a, x$ are real numbers, $| a |$ $< 1$ , $| x |$ $< 1$ , then $1 + (1 + a) x + (1 + a + a^{2}) x^{2} + \dots \infty$ is equal to

$\frac{1}{(1 - a) (1 - a x)}$
$\frac{1}{(1 - a) (1 - x)}$
$\frac{1}{(1 - x) (1 - a x)}$
$\frac{1}{(1 - a x) (1 + a)}$

1

2

3

4

(3)

$1 + (1 + a) x + (1 + a + a^{2}) x^{2} + \dots to \infty$

$= \frac{1}{1 - a} [(1 - a) + (1 - a^{2}) x + (1 - a^{3}) x^{2} + \dots to \infty]$

$= \frac{1}{1 - a} [(1 + x + x^{2} + \dots to \infty) - (a + a^{2} x + a^{3} x^{2} + \dots to \infty)]$

$= \frac{1}{1 - a} [\frac{1}{1 - x} - \frac{a}{1 - a x}] = \frac{1}{(1 - x) (1 - a x)}$

Q 38 :

How many digits will be there after decimal point in 12th term of the sequence: 32, 16, 8, 4, ..... ?

5
6
7
8

1

2

3

4

(2)

$We have, 32, 16, 8, 4, \dots are in G.P.$

$a = 32, r = \frac{1}{2}$

Then, $12^{th}$ term of the sequence $= a r^{12 - 1}$

$= 32 \times \frac{1}{2^{11}} = 2^{5} \times 2^{- 11} = 2^{- 6} = 0.015625$

Hence, six digits will be there after decimal point in $12^{th}$ term of the sequence.

Q 39 :

If three distinct numbers $a, b, c$ are in G.P. and the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?

$d, e, f are in A.P.$
$\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in G.P.$
$d, e, f are in G.P.$
$\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in A.P.$

1

2

3

4

(4)

$Given a, b, c are in G.P.$

Let $a = a,$ $b = a r$ and $c = a r^{2}$ ...(i)

Now, consider $a x^{2} + 2 b x + c = 0$

$∴$ $a x^{2} + 2 a r x + a r^{2} = 0$ $[Using (i)]$

$\Rightarrow$ $x^{2} + 2 r x + r^{2} = 0 \Rightarrow {(x + r)}^{2} = 0 \Rightarrow x = - r$

Since, $x = - r$ is also a root of $d x^{2} + 2 e x + f = 0$

$∴$ $d r^{2} - 2 e r + f = 0$

$\Rightarrow$ $d (\frac{c}{a}) - 2 e (\frac{c}{b}) + f = 0$ $[∵ (- r) (- r) = \frac{c}{a} and \frac{a r^{2}}{a r} = \frac{c}{b}]$

$\Rightarrow \frac{d c}{a} + f = \frac{2 e c}{b}$

$\Rightarrow \frac{d}{a} + \frac{f}{c} = \frac{2 e}{b}$

$∴$ $\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in A.P.$

Q 40 :

If $\frac{a^{n} + b^{n}}{a^{n - 1} + b^{n - 1}}$ is a G.M. between $a$ and $b$ , then the value of $n$ is

$\frac{1}{5}$
$\frac{1}{4}$
$\frac{1}{3}$
$\frac{1}{2}$

1

2

3

4

(4)

$Given, \frac{a^{n} + b^{n}}{a^{n - 1} + b^{n - 1}} = \sqrt{a b}$

$\Rightarrow$ $\frac{a^{n} + b^{n}}{\sqrt{a b}} = a^{n - 1} + b^{n - 1}$

$\Rightarrow$ $\frac{a^{n - \frac{1}{2}}}{\sqrt{b}} + \frac{b^{n - \frac{1}{2}}}{\sqrt{a}} - a^{n - 1} - b^{n - 1} = 0$

$\Rightarrow$ $a^{n - \frac{1}{2}} (b^{- \frac{1}{2}} - a^{- \frac{1}{2}}) + b^{n - \frac{1}{2}} (a^{- \frac{1}{2}} - b^{- \frac{1}{2}}) = 0$

$\Rightarrow$ $(b^{- \frac{1}{2}} - a^{- \frac{1}{2}}) (a^{n - \frac{1}{2}} - b^{n - \frac{1}{2}}) = 0$

$\Rightarrow$ $b^{- \frac{1}{2}} - a^{- \frac{1}{2}} = 0 \Rightarrow \sqrt{a} = \sqrt{b}$

or $a^{n - \frac{1}{2}} - b^{n - \frac{1}{2}} = 0$

$\Rightarrow$ ${(\frac{a}{b})}^{n - \frac{1}{2}} = 1 = {(\frac{a}{b})}^{0} \Rightarrow n = \frac{1}{2}$

Q 41 :

Two numbers $x$ and $y$ have arithmetic mean 9 and geometric mean 4. Then $x$ and $y$ are the roots of

$x^{2} - 18 x - 16 = 0$
$x^{2} - 18 x + 16 = 0$
$x^{2} + 18 x - 16 = 0$
$x^{2} + 18 x + 16 = 0$

1

2

3

4

(2)

$Given, arithmetic mean of x and y is 9 .$

$i.e., \frac{x + y}{2} = 9 \Rightarrow x + y = 18$

Geometric mean of $x$ and $y$ is 4, i.e., $\sqrt{x y} = 4 \Rightarrow x y = 16$

Now, sum of roots $(x + y) = 18$

Product of roots $(x y) = 16$

$∴$ Required quadratic equation is $x^{2} - 18 x + 16 = 0$

Q 42 :

Find the natural number $' a'$ for which $\sum_{k = 1}^{n} f (a + k) = 16 (2^{n} - 1),$ where $f (x) = 2^{x}$ .

(3)

$f (x) = 2^{x} for all x \in N$

$∴$ $\sum_{k = 1}^{n} f (a + k) = 16 (2^{n} - 1) \Rightarrow \sum_{k = 1}^{n} 2^{a + k} = 16 (2^{n} - 1)$

$\Rightarrow \sum_{k = 1}^{n} 2^{a} \cdot 2^{k} = 16 (2^{n} - 1) \Rightarrow 2^{a} (\sum_{k = 1}^{n} 2^{k}) = 16 (2^{n} - 1)$

$\Rightarrow 2^{a} (2 + 2^{2} + \dots + 2^{n}) = 16 (2^{n} - 1)$

$\Rightarrow 2^{a} {2 [\frac{2^{n} - 1}{2 - 1}]} = 16 (2^{n} - 1)$

$\Rightarrow 2^{a + 1} (2^{n} - 1) = 16 (2^{n} - 1)$

$\Rightarrow 2^{a + 1} = 2^{4} \Rightarrow a + 1 = 4 \Rightarrow a = 3$

Q 43 :

If $a_{1}, a_{2}, a_{3}, \dots, a_{20}$ are in A.P. and $a_{1} + a_{20} = 45$ , then $a_{1} + a_{2} + a_{3} + \dots + a_{20}$ is equal to _________.

(450)

$S_{n} = \frac{n}{2} (a + l)$

$∴$ $S_{20} = \frac{20}{2} (45) = 450$ $(∵ a_{1} + a_{20} = 45)$

Q 44 :

Let $a_{1}, a_{2}, a_{3}, \dots, a_{n}, \dots$ be in A.P. If $a_{3} + a_{7} + a_{11} + a_{15} = 72,$ then the sum of its first 17 terms is equal to _______.

(306)

$We have, a_{3} + a_{7} + a_{11} + a_{15} = 72$

$\Rightarrow$ $(a_{3} + a_{15}) + (a_{7} + a_{11}) = 72$

$Now, a_{3} + a_{15} = a_{7} + a_{11} = a_{1} + a_{17}$ $∴$ $a_{1} + a_{17} = 36$

$Now, S_{17} = \frac{17}{2} [a_{1} + a_{17}] = 17 \times 18 = 306$

Q 45 :

If 25th term of an A.P. is 15 and if its 15th term is 25, then the 40th term of the A.P. is ______.

(0)

Let $a$ be the first term and $d$ be the common difference.

According to question, $T_{25} = 15 \Rightarrow a + 24 d = 15$ ...(i)

$T_{15} = 25 \Rightarrow a + 14 d = 25$ ...(ii)

On solving (i) and (ii), we get $a = 39,$ $d = - 1$

So, $T_{40} = a + 39 d = 39 + 39 (- 1) = 0$

Q 46 :

Let the sum of the first three terms of an A.P. be 39 and the sum of its last four terms be 178. If the first term of this A.P. is 10, then the median of the A.P. is ______.

(29.5)

$Let a, a + d, a + 2 d be first three terms of an A.P.$

$So, a + a + d + a + 2 d = 39 \Rightarrow d = 3$ $(∵ a = 10)$

$Sum of last four terms = 178$

$\Rightarrow l - 3 d + l - 2 d + l - d + l = 178 \Rightarrow l = 49$

$A.P. is 10, 13, 16, 19, \dots, 46, 49 .$

$Median = \frac{10 + 49}{2} = 29.5$

Q 47 :

The sum of the series $1 \cdot n + 2 \cdot (n - 1) + 3 \cdot (n - 2) + \dots + n \cdot 1$ is

$\frac{n (n + 1) (n + 2)}{6}$
$\frac{n (n + 1) (n + 2)}{3}$
$\frac{n (n + 1) (2 n + 1)}{6}$
$\frac{n (n + 1) (2 n + 1)}{3}$

1

2

3

4

(1)

$\sum_{r = 1}^{n} r (n - r + 1) = \sum_{r = 1}^{n} (n + 1) r - \sum_{r = 1}^{n} r^{2}$

$= (n + 1) \sum n - \sum n^{2} = \frac{{(n + 1)}^{2} n}{2} - \frac{n (n + 1) (2 n + 1)}{6}$

$= \frac{n (n + 1)}{6} (3 n + 3 - 2 n - 1) = \frac{n (n + 1) (n + 2)}{6}$

Topic Question Set

Q 21 :

The interior angles of a polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number of sides of the polygon is

Q 22 :

The sum to infinity of the series $1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \frac{14}{3^{4}} + \dots$ is

Q 23 :

In a G.P., $t_{2} + t_{5} = 216$ and $t_{4} : t_{6} = 1 : 4$ and all terms are integers, then its first term is

Q 24 :

If $3^{rd}$ , $7^{th}$ and $12^{th}$ terms of an A.P. are three consecutive terms of a G.P., then the common ratio of the G.P. is

Q 25 :

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

Q 26 :

If ${(10)}^{9} + 2 {(11)}^{1} {(10)}^{8} + 3 {(11)}^{2} {(10)}^{7} + \dots + 10 {(11)}^{9} = k {(10)}^{9}$ , then $k$ is equal to

Q 27 :

If $a, b, c$ are in H.P., then $\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}$ will be in

Q 28 :

If A, G and H are respectively the A.M., the G.M. and the H.M. between two positive numbers $a$ and $b$ , then the correct relation is

Q 29 :

The positive numbers $a_{1}, a_{2}, a_{3}$ are in A.P. such that $a_{1} a_{2} a_{3} = 64$ . Then the minimum value of $a_{2}$ is

Q 30 :

If A.M. and G.M. of $x$ and $y$ are in the ratio $p : q$ , then $x : y$ is

Q 31 :

The sum of the first 8 terms of $\frac{1^{2}}{2} + \frac{1^{2} + 2^{2}}{3} + \frac{1^{2} + 2^{2} + 3^{2}}{4} + \dots$ is

Q 32 :

If the $10^{th}$ term of an A.P. is $\frac{1}{20}$ and its $20^{th}$ term is $\frac{1}{10}$ , then the sum of its first 200 terms is

Q 33 :

If the $19^{th}$ term of a non-zero A.P. is zero, then its ( $49^{th}$ term) : ( $29^{th}$ term) is

Q 34 :

The sum of all those terms, of the arithmetic progression 3, 8, 13, ....., 373, which are not divisible by 3, is equal to ______.

Q 35 :

Let the digits $a, b, c$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Q 36 :

The sum of the common terms of the following three arithmetic progressions 3, 7, 11, 15, ...., 399; 2, 5, 8, 11, ...., 359 and 2, 7, 12, 17, ...., 197, is equal to ______.

Q 37 :

If $a, x$ are real numbers, $| a |$ $< 1$ , $| x |$ $< 1$ , then $1 + (1 + a) x + (1 + a + a^{2}) x^{2} + \dots \infty$ is equal to

Q 38 :

How many digits will be there after decimal point in 12th term of the sequence: 32, 16, 8, 4, ..... ?

Q 39 :

If three distinct numbers $a, b, c$ are in G.P. and the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?

Q 40 :

If $\frac{a^{n} + b^{n}}{a^{n - 1} + b^{n - 1}}$ is a G.M. between $a$ and $b$ , then the value of $n$ is

Q 41 :

Two numbers $x$ and $y$ have arithmetic mean 9 and geometric mean 4. Then $x$ and $y$ are the roots of

Q 42 :

Find the natural number $' a'$ for which $\sum_{k = 1}^{n} f (a + k) = 16 (2^{n} - 1),$ where $f (x) = 2^{x}$ .

Q 43 :

If $a_{1}, a_{2}, a_{3}, \dots, a_{20}$ are in A.P. and $a_{1} + a_{20} = 45$ , then $a_{1} + a_{2} + a_{3} + \dots + a_{20}$ is equal to _________.

Q 44 :

Let $a_{1}, a_{2}, a_{3}, \dots, a_{n}, \dots$ be in A.P. If $a_{3} + a_{7} + a_{11} + a_{15} = 72,$ then the sum of its first 17 terms is equal to _______.

Q 45 :

If 25th term of an A.P. is 15 and if its 15th term is 25, then the 40th term of the A.P. is ______.

Q 46 :

Let the sum of the first three terms of an A.P. be 39 and the sum of its last four terms be 178. If the first term of this A.P. is 10, then the median of the A.P. is ______.

Q 47 :

The sum of the series $1 \cdot n + 2 \cdot (n - 1) + 3 \cdot (n - 2) + \dots + n \cdot 1$ is

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Topic Question Set

Q 21 : The interior angles of a polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number of sides of the polygon is

Q 22 : The sum to infinity of the series 1+23+632+1033+1434+⋯ is

Q 23 : In a G.P., t2+t5=216 and t4:t6=1:4 and all terms are integers, then its first term is

Q 24 : If 3rd, 7th and 12th terms of an A.P. are three consecutive terms of a G.P., then the common ratio of the G.P. is

Q 25 : Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

Q 26 : If (10)9+2(11)1(10)8+3(11)2(10)7+⋯+10(11)9=k(10)9, then k is equal to

Q 27 : If a,b,c are in H.P., then ab+c, bc+a, ca+b will be in

Q 28 : If A, G and H are respectively the A.M., the G.M. and the H.M. between two positive numbers a and b, then the correct relation is

Q 29 : The positive numbers a1,a2,a3 are in A.P. such that a1a2a3=64. Then the minimum value of a2 is

Q 30 : If A.M. and G.M. of x and y are in the ratio p:q, then x:y is

Q 31 : The sum of the first 8 terms of 122+12+223+12+22+324+⋯ is

Q 32 : If the 10th term of an A.P. is 120 and its 20th term is 110, then the sum of its first 200 terms is

Q 33 : If the 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is

Q 34 : The sum of all those terms, of the arithmetic progression 3, 8, 13, ....., 373, which are not divisible by 3, is equal to ______.

Q 35 : Let the digits a, b, c be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Q 36 : The sum of the common terms of the following three arithmetic progressions 3, 7, 11, 15, ...., 399; 2, 5, 8, 11, ...., 359 and 2, 7, 12, 17, ...., 197, is equal to ______.

Q 37 : If a,x are real numbers, |a|<1, |x|<1, then 1+(1+a)x+(1+a+a2)x2+⋯∞ is equal to

Q 38 : How many digits will be there after decimal point in 12th term of the sequence: 32, 16, 8, 4, ..... ?

Q 39 : If three distinct numbers a,b,c are in G.P. and the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root, then which one of the following statements is correct?

Q 40 : If an+bnan-1+bn-1 is a G.M. between a and b, then the value of n is

Q 41 : Two numbers x and y have arithmetic mean 9 and geometric mean 4. Then x and y are the roots of

Q 42 : Find the natural number 'a' for which ∑k=1nf(a+k)=16(2n-1), where f(x)=2x.

Q 43 : If a1,a2,a3,…,a20 are in A.P. and a1+a20=45, then a1+a2+a3+⋯+a20 is equal to _________.

Q 44 : Let a1,a2,a3,…,an,… be in A.P. If a3+a7+a11+a15=72, then the sum of its first 17 terms is equal to _______.

Q 45 : If 25th term of an A.P. is 15 and if its 15th term is 25, then the 40th term of the A.P. is ______.

Q 46 : Let the sum of the first three terms of an A.P. be 39 and the sum of its last four terms be 178. If the first term of this A.P. is 10, then the median of the A.P. is ______.

Q 47 : The sum of the series 1·n+2·(n-1)+3·(n-2)+⋯+n·1 is

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Q 21 :

The interior angles of a polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number of sides of the polygon is

Q 22 :

The sum to infinity of the series $1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \frac{14}{3^{4}} + \dots$ is

Q 23 :

In a G.P., $t_{2} + t_{5} = 216$ and $t_{4} : t_{6} = 1 : 4$ and all terms are integers, then its first term is

Q 24 :

If $3^{rd}$ , $7^{th}$ and $12^{th}$ terms of an A.P. are three consecutive terms of a G.P., then the common ratio of the G.P. is

Q 25 :

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

Q 26 :

If ${(10)}^{9} + 2 {(11)}^{1} {(10)}^{8} + 3 {(11)}^{2} {(10)}^{7} + \dots + 10 {(11)}^{9} = k {(10)}^{9}$ , then $k$ is equal to

Q 27 :

If $a, b, c$ are in H.P., then $\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}$ will be in

Q 28 :

If A, G and H are respectively the A.M., the G.M. and the H.M. between two positive numbers $a$ and $b$ , then the correct relation is

Q 29 :

The positive numbers $a_{1}, a_{2}, a_{3}$ are in A.P. such that $a_{1} a_{2} a_{3} = 64$ . Then the minimum value of $a_{2}$ is

Q 30 :

If A.M. and G.M. of $x$ and $y$ are in the ratio $p : q$ , then $x : y$ is

Q 31 :

The sum of the first 8 terms of $\frac{1^{2}}{2} + \frac{1^{2} + 2^{2}}{3} + \frac{1^{2} + 2^{2} + 3^{2}}{4} + \dots$ is

Q 32 :

If the $10^{th}$ term of an A.P. is $\frac{1}{20}$ and its $20^{th}$ term is $\frac{1}{10}$ , then the sum of its first 200 terms is

Q 33 :

If the $19^{th}$ term of a non-zero A.P. is zero, then its ( $49^{th}$ term) : ( $29^{th}$ term) is

Q 34 :

The sum of all those terms, of the arithmetic progression 3, 8, 13, ....., 373, which are not divisible by 3, is equal to ______.

Q 35 :

Let the digits $a, b, c$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Q 36 :

The sum of the common terms of the following three arithmetic progressions 3, 7, 11, 15, ...., 399; 2, 5, 8, 11, ...., 359 and 2, 7, 12, 17, ...., 197, is equal to ______.

Q 37 :

If $a, x$ are real numbers, $| a |$ $< 1$ , $| x |$ $< 1$ , then $1 + (1 + a) x + (1 + a + a^{2}) x^{2} + \dots \infty$ is equal to

Q 38 :

How many digits will be there after decimal point in 12th term of the sequence: 32, 16, 8, 4, ..... ?

Q 39 :

If three distinct numbers $a, b, c$ are in G.P. and the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?

Q 40 :

If $\frac{a^{n} + b^{n}}{a^{n - 1} + b^{n - 1}}$ is a G.M. between $a$ and $b$ , then the value of $n$ is

Q 41 :

Two numbers $x$ and $y$ have arithmetic mean 9 and geometric mean 4. Then $x$ and $y$ are the roots of

Q 42 :

Find the natural number $' a'$ for which $\sum_{k = 1}^{n} f (a + k) = 16 (2^{n} - 1),$ where $f (x) = 2^{x}$ .

Q 43 :

If $a_{1}, a_{2}, a_{3}, \dots, a_{20}$ are in A.P. and $a_{1} + a_{20} = 45$ , then $a_{1} + a_{2} + a_{3} + \dots + a_{20}$ is equal to _________.

Q 44 :

Let $a_{1}, a_{2}, a_{3}, \dots, a_{n}, \dots$ be in A.P. If $a_{3} + a_{7} + a_{11} + a_{15} = 72,$ then the sum of its first 17 terms is equal to _______.

Q 45 :

If 25th term of an A.P. is 15 and if its 15th term is 25, then the 40th term of the A.P. is ______.

Q 46 :

Let the sum of the first three terms of an A.P. be 39 and the sum of its last four terms be 178. If the first term of this A.P. is 10, then the median of the A.P. is ______.

Q 47 :

The sum of the series $1 \cdot n + 2 \cdot (n - 1) + 3 \cdot (n - 2) + \dots + n \cdot 1$ is