Sequences & Series — JEE Maths Practice Questions & Solutions

mediumPYQ · JEE Main 2026

Let

\alpha, \beta

be the roots of

x^2 - x + p = 0

and

\gamma, \delta

be the roots of

x^2 - 4x + q = 0

, where

p, q \in \mathbb{Z}

. If

\alpha, \beta, \gamma, \delta

are in G.P., then

|p + q|

equals:

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mediumPYQ · JEE Main 2026

If the sum of the first 10 terms of the series

\dfrac{1}{1+1^4\cdot4}+\dfrac{2}{1+2^4\cdot4}+\dfrac{3}{1+3^4\cdot4}+\cdots

is

\dfrac{m}{n}

with

\gcd(m,n)=1

, then

m+n

is equal to:

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mediumPYQ · JEE Main 2026

Let

A

be the set of first

101

terms of an A.P. whose first term is

1

and common difference is

5

, and let

B

be the set of first

71

terms of an A.P. whose first term is

9

and common difference is

7

. Then the number of elements in

A\cap B

which are divisible by

3

is

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mediumPYQ · JEE Main 2026

Let

A_1, A_2, \ldots, A_{39}

be 39 arithmetic means between 59 and 159. Then the mean of

A_{25}

,

A_{28}

,

A_{31}

and

A_{36}

is equal to:

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mediumPYQ · JEE Main 2026

The sum

\dfrac{1^3}{1}+\dfrac{1^3+2^3}{1+3}+\dfrac{1^3+2^3+3^3}{1+3+5}+\cdots

up to 8 terms, is

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mediumPYQ · JEE Main 2026

Let

a_1,a_2,a_3,\ldots

be an A.P. and

g_1=a_1,g_2,g_3,\ldots

be an increasing G.P. If

a_1=a_2+g_2=1

and

a_3+g_3=4

, then

a_{10}+g_5

is equal to

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mediumPYQ · JEE Main 2026

Let

\alpha=\dfrac14+\dfrac18+\dfrac1{16}+\cdots\infty

and

\beta=\dfrac13+\dfrac19+\dfrac1{27}+\cdots\infty

. Then the value of

(0.2)^{\log_{\sqrt5}(\alpha)}+(0.04)^{\log_5(\beta)}

is

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mediumPYQ · JEE Main 2026

If

\displaystyle\sum_{k=1}^{n}a_k=6n^3

, then

\displaystyle\sum_{k=1}^{6}\left(\dfrac{a_{k+1}-a_k}{36}\right)^2

is equal to

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mediumPYQ · JEE Main 2026

The value of

1^3-2^3+3^3-\cdots+15^3

is

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mediumPYQ · JEE Main 2026

The sum of the first ten terms of an A.P. is

160

and the sum of the first two terms of a G.P. is

8

. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to the common difference of the A.P., then the sum of all possible values of the first term of the G.P. is

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easyPYQ · JEE Main 2026

\displaystyle\sum_{n=1}^{10}\left(\dfrac{528}{n(n+1)(n+2)}\right)

is equal to

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easyPYQ · JEE Main 2026

Let the sum of the first

n

terms of an A.P. be

3n^2+5n

. Then the sum of squares of the first

10

terms of the A.P. is

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easyPYQ · JEE Main 2026

The first term of an A.P. of

30

non-negative terms is

\dfrac{10}{3}

. If the sum of this A.P. is the cube of its last term, then its common difference is

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expert

If

(1 + 3 + 5 + \cdots + a) + (1 + 3 + 5 + \cdots + b) = (1 + 3 + 5 + \cdots + c),

such that (i)

a + b + c = 57

and (ii)

a < 10

, then find the value of

(2b - 2c + a)

.

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hard

The value of

\displaystyle\sum_{n=1}^{2008}\frac{\sqrt{n^4+2n^3+3n^2+2n+1}}{n(n+1)}

is

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hard

If positive numbers

x

,

y

,

z

are in A.P., then the minimum value of

\frac{x+y}{2y-x} + \frac{y+z}{2y-z}

is less than

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hard

Match Column-I with Column-II.

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hard

Let

A_i

(i = 1, 2, \ldots, 6)

represent

6

arithmetic means,

G_i

(i = 1, 2, \ldots, 6)

represent

6

geometric means and

H_i

(i = 1, 2, \ldots, 6)

represent

6

harmonic means between the two numbers

2

and

72

, and let

G

represent the single geometric mean between

2

and

72

. Then the value of

\frac{\left(\displaystyle\prod_{i=1}^{6} G_i\right)\left(\displaystyle\sum_{i=1}^{6} \frac{1}{H_i}\right)} {\displaystyle\sum_{i=1}^{6} A_i}

is

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hard

If

1^{2} + 2\cdot 2^{2} + 3^{2} + 2\cdot 4^{2} + 5^{2} + 2\cdot 6^{2} + \cdots

up to

n

terms is equal to

1575

, then find the value of

n

.

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hard

If

1 + \frac{(1+2)(1+3)}{2^{2}\left(1^{3}+2^{3}\right)} + \frac{(1+2+3)(1+3+5)}{3^{2}\left(1^{3}+2^{3}+3^{3}\right)} + \frac{(1+2+3+4)(1+3+5+7)}{4^{2}\left(1^{3}+2^{3}+3^{3}+4^{3}\right)} + \cdots

up to

10

terms is equal to

\dfrac{a}{b}

, where

a

and

b

are relatively prime, then

a - b

is equal to

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medium

Let

a_1, a_2, \ldots, a_{4001}

be in A.P. such that

\frac{1}{a_1 a_2}+\frac{1}{a_2 a_3}+\cdots+\frac{1}{a_{4000} a_{4001}}=10

and

a_2+a_{4000}=50

. Then

|a_1-a_{4001}|

equals

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medium

If

\frac{p+q\cdot 5^x}{p-q\cdot 5^x}=\frac{q+r\cdot 5^x}{q-r\cdot 5^x}=\frac{r+s\cdot 5^x}{r-s\cdot 5^x}

, then

p, q, r, s

are in

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medium

A sequence

A_n

is defined by

A_1=\frac12

and

A_n=\frac{2n-3}{2n}A_{n-1}

for

n\ge 2

. Then

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medium

The sequence

\{x_n\}

is defined by

x_{k+1}=x_k^2+x_k

and

x_1=\frac12

. Then

\left[\frac{1}{x_1+1}+\frac{1}{x_2+1}+\cdots+\frac{1}{x_{100}+1}\right]

, where

[\,\cdot\,]

is the greatest integer function, equals

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medium

If

a, b, c, d

are positive reals with

abcd=1

, then the minimum value of

(1+a)(1+b)(1+c)(1+d)

is

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medium

Let

a_1, a_2, a_3, a_4, a_5

be five terms of a geometric sequence satisfying

0<a_1<a_2<a_3<a_4<a_5<100

where each term is an integer. Then the number of such geometric progressions is

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medium

If

a_1, a_2, \ldots, a_n

are distinct odd natural numbers not divisible by any prime greater than

5

, then

\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}

is less than

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medium

If

a, b, c, d

are in G.P., then

(a^2+b^2+c^2)(b^2+c^2+d^2)

equals

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medium

If

a_i>0

for

1\le i\le 5

and

a_1 a_2 a_3 a_4 a_5=6

, then the least value of

a_1+2a_2+3a_3+4a_4+5a_5

is

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medium

If

a, b, c

are positive with

a+b+c=6

, then the minimum value of

\left(a+\frac1b\right)^2+\left(b+\frac1c\right)^2+\left(c+\frac1a\right)^2

is

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medium

The sum of the series

\frac{4}{19}+\frac{44}{19^2}+\frac{444}{19^3}+\cdots

to infinity is

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medium

If

a+b+c+d+e=8

and

a^2+b^2+c^2+d^2+e^2=16

, where

a,b,c,d,e

are non-negative reals and the range of

e

is

\left[\ell,\frac{m}{n}\right]

with

\gcd(m,n)=1

, then

\ell+m+n

is

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medium

If

A, G, H

are respectively the A.M., G.M. and H.M. between two positive numbers, and

xA=yG=zH

where

x, y, z

are non-zero positive quantities, then

x, y, z

are in

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medium

The

2008

th term of the sequence

1,\ 2,2,2,\ 3,3,3,3,3,3,\ 4,\ldots

where

n

occurs

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

times, equals

View solution →

medium

If

H_1, H_2, \ldots, H_{20}

are

20

harmonic means between

2

and

3

, then

\frac{H_1+2}{H_1-2}+\frac{H_{20}+3}{H_{20}-3}=

View solution →

medium

If

\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots=\frac{\pi^2}{6}

, then

\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots

is equal to

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medium

In a sequence of

(4n+1)

terms, the first

(2n+1)

terms are in A.P. with common difference

2

, and the last

(2n+1)

terms are in G.P. with common ratio

0.5

. If the middle terms of the A.P. and the G.P. are equal, then the middle term of the sequence is

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medium

The sum of the series

\frac{x}{1-x^2}+\frac{x^2}{1-x^4}+\frac{x^4}{1-x^8}+\cdots

to infinity, for

|x|<1

, is

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medium

If

n

arithmetic means are inserted between

a

and

2b

, and also between

2a

and

b

(

a,b\in\mathbb{R}

), and the

m

th means of the two sets are equal, then

a:b

equals

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medium

If

a, b, c

are in A.P.,

p, p'

are the A.M. and G.M. between

a

and

b

, and

q, q'

are the A.M. and G.M. between

b

and

c

, then

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medium

Through the centroid of an equilateral triangle a line parallel to the base is drawn. On this line an arbitrary interior point

P

is taken. Let

h

be the distance of

P

from the base, and

h_1, h_2

its distances from the other two sides. Then

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medium

If the common ratio of a geometric progression is positive, then the maximum value of the ratio of the second term to the sum of the first three terms equals

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medium

The minimum value of

\frac{12}{x} + \frac{18}{y} + xy ,

where

x

and

y

are positive real numbers, is

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medium

The largest term of the sequence

\frac{1}{503},\frac{4}{524},\frac{9}{581},\frac{16}{692},\ldots

is

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medium

In a triangle

ABC

,

\sin^{2}A+\sin^{2}B=\sin^{2}C

and

\sin A,\ \sin B,\ \sin C

are in A.P. in that order. If the inradius of the triangle is

2

, then which of the following is/are correct?

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easy

a, b, c, d

are in increasing G.P. If the A.M. between

a

and

b

is

6

and the A.M. between

c

and

d

is

54

, then the A.M. of

a

and

d

is

View solution →

easy

Let

a_1, a_2, \ldots

and

b_1, b_2, \ldots

be arithmetic progressions with

a_1=25

,

b_1=75

and

a_{100}+b_{100}=100

. Then the sum of the first hundred terms of

a_1+b_1,\ a_2+b_2,\ \ldots

is

View solution →

easy

If

S_n=\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+\cdots+\frac{1}{n(n+1)(n+2)}

, then

8S_\infty

is

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easy

If the sum to infinity of

1+4x+7x^2+10x^3+\cdots

is

\frac{35}{16}

, where

|x|<1

, then

x

equals

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easy

If

t_r=\frac{1^2+2^2+\cdots+r^2}{1^3+2^3+\cdots+r^3}

and

S_n=\sum_{r=1}^{n}(-1)^r t_r

, then

\lim_{n\to\infty}S_n

equals

View solution →

easy

If

S_n=\frac{1\cdot2}{3!}+\frac{2\cdot2^2}{4!}+\frac{3\cdot2^3}{5!}+\cdots

up to

n

terms, then the sum to infinite terms is

View solution →

easy

The sum of the first

n

terms (

n>1

) of an A.P. is

153

and the common difference is

2

. If the first term is an integer, the number of possible values of

n

is

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easy

If the lengths of the sides of a right triangle are in A.P., then the sines of the acute angles are

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easy

Let

f(n), g(n)

denote the sums of

n

terms of the sequences

2,5,10,17,26,\ldots

and

2,6,12,20,\ldots

respectively. Then

\lim_{n\to\infty}\frac{f(n)}{g(n)}

is

View solution →

easy

Consider

\alpha_0, \alpha_1, \alpha_2, \ldots

with

\alpha_0=17.23

,

\alpha_1=33.23

and

\alpha_{r+2}=\frac{\alpha_r+\alpha_{r+1}}{2}

for all

r\ge 0

. The value of

|\alpha_{10}-\alpha_9|

is

View solution →

easy

If

1-\frac13+\frac15-\frac17+\frac19-\frac{1}{11}+\cdots=\frac{\pi}{4}

, then the value of

\frac{1}{1\cdot3}+\frac{1}{5\cdot7}+\frac{1}{9\cdot11}+\cdots

is

View solution →

easy

If

a, b, c

are three positive numbers, then the minimum value of

\frac{ab(a+b)+bc(b+c)+ca(c+a)}{abc}

is

View solution →

easy

If the sum of

n

terms of an A.P. is

cn(n-1)

with

c\ne 0

, then the sum of the squares of these terms is

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easy

If

a, b, c

are three distinct numbers such that

a, b, c

are in A.P. and

b-a,\ c-b,\ a

are in G.P., then

a:b:c

is

View solution →

easy

If

a_k=\frac{1}{k(k+1)}

for

k=1,2,\ldots,n

, then

\left(\sum_{k=1}^{n}a_k\right)^2

is equal to

View solution →

easy

The value of the series is

\displaystyle\sum_{k=1}^{\infty}\frac{1}{k!}\left(\sum_{n=1}^{k}2^{n-1}\right)

View solution →

easy

If

1,\ \log_9\sqrt{3^x+48},\ \log_9\left(3^x-\frac83\right)

are in A.P., then the value of

x

is

View solution →

easy

If

a, b, c

are in H.P., then

\frac{a}{b+c},\ \frac{b}{c+a},\ \frac{c}{a+b}

are in

View solution →

easy

The ratio of the sum of the first three terms of a G.P. to the sum of the first six terms is

64:91

. The common ratio of the G.P. is

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easy

The sum of the series

3+5+9+17+33+\cdots

to

n

terms is

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easy

If one A.M.

A

and two G.M.s

p

and

q

are inserted between two numbers

a

and

b

, then which is true?

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easy

If the fourth term of a G.P. is

3

, then the product of the first seven terms is

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Sequences & Series — JEE Maths Practice Questions & Solutions

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