Let a 1 , a 2 , a 3 , … , a 100 be an arithmetic progression with a 1 = 3 and S p = ∑ i = 1 p a i , 1 ≤ p ≤ 100 for any integer n with 1 ≤ n ≤ 20 , let m = 5 n . If S m S n does not depend on n , then a 2 is

Sum of n terms of AP is given by n 2 2 a + n - 1 d Given m = 5 n S m S n = S 5 n S n = 5 n 2 6 + 5 n - 1 d n 2 6 + n - 1 d = 5 6 + 5 n - 1 d 6 + n - 1 d Since S m S n is independent of n So it can be possible when 6 - d = 0 d = 6 a 2 = a 1 + d = 3 + 6 = 9 answer And can also possible when d = 0 Then a 2 = a 1 + d ⇒ 3 + 0 ⇒ 3 answer

Let a 1 , a 2 , a 3 , … , a 100 be an arithmetic progression with a 1 = 3 and S p = ∑ i = 1 p a i , 1 ≤ p ≤ 100 for any integer n with 1 ≤ n ≤ 20 , let m = 5 n . If S m S n does not depend on n , then a 2 is

Sum of n terms of AP is given by n 2 2 a + n - 1 d Given m = 5 n S m S n = S 5 n S n = 5 n 2 6 + 5 n - 1 d n 2 6 + n - 1 d = 5 6 + 5 n - 1 d 6 + n - 1 d Since S m S n is independent of n So it can be possible when 6 - d = 0 d = 6 a 2 = a 1 + d = 3 + 6 = 9 answer And can also possible when d = 0 Then a 2 = a 1 + d ⇒ 3 + 0 ⇒ 3 answer

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 9

HARD

JEE Main/Advance

IMPORTANT

Earn 100

Let $a_{1}, a_{2}, a_{3}, \dots, a_{100}$ be an arithmetic progression with $a_{1} = 3$ and $S_{p} = \sum_{i = 1}^{p} a_{i}, 1 \leq p \leq 100$ for any integer $n$ with $1 \leq n \leq 20,$ let $m = 5 n .$ If $\frac{S_{m}}{S_{n}}$ does not depend on $n,$ then $a_{2}$ is

Important Questions on Sequences and Series

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 10

The minimum value of the sum of real numbers

a^{- 5}, a^{- 4}, 3 a^{- 3}, 1, a^{8}

and

a^{10}

, where

a > 0

is

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 11

Let

a_{1}, a_{2}, a_{3}, \dots .

be in harmonic progression with

a_{1} = 5

and

a_{20} = 25

. The least positive integer

n

for which

a_{n} < 0

is

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 12

Let

S_{n} = \sum_{k = 1}^{4 n} (- 1)^{\frac{k (k + 1)}{2}} k^{2} .

Then

S_{n}

can take value

(s)

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 13

A pack contains

n

cards numbered from

1

to

n .

Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is

1224 .

If the smaller of the numbers on the removed cards is

k,

then

k - 20 =

.

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 15

Suppose that all the terms of an arithmetic progression (

A . P .

) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is

6 : 11

and the seventh term lies in between

130

and

140

, then the common difference of this

A . P .

is

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 16

Let

a_{1}, a_{2}, a_{3}, \dots

be terms of an

AP

. If

\frac{a_{1} + a_{2} + \dots + a_{p}}{a_{1} + a_{2} + \dots + a_{q}} = \frac{p^{2}}{q^{2}}, p \neq q

, then

\frac{a_{6}}{a_{21}}

equals

MEDIUM

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 17

If

a_{1}, a_{2}, \dots, a_{n}

are in

HP

, then the expression

a_{1} a_{2} + a_{2} a_{3} + \dots + a_{n - 1} a_{n}

is equal to

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 18

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then, the common ratio of this progression equals