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_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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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

Consider

ax^2-bx+c=0

with

a, b, c\in\mathbb{N}

. If it has two distinct real roots in

(1,2)

, then

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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

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

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

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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}=

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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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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

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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

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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

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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

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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

If a right triangle has sides in AP, solving Pythagoras gives ratio 3:4:5, so the acute angle sines are 3/5 and 4/5

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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

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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

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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

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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

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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

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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

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easy

The value of the series is

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

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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

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easy

If

a, b, c

are in H.P., then

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

are in

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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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