The sum of the series ${1^2} + 1 + {2^2} + 2 + {3^2} + 3 + \ldots + {n^2} + n$ is

$\frac{{n\left( {n + 1} \right)}}{2}$

${\left\{ {\frac{{n\left( {n + 1} \right)}}{2}} \right\}^2}$

$\frac{{n\left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right)}}{4}$

Given that $n$ AM’s are inserted between two sets of numbers $a,\;2b\;{\rm{and}}\;2a,\;b\;{\rm{where}}\;a,\;b \in R.$. Suppose further that $m$th mean between these sets of numbers is same, then the ratio $a\;:b$ equals

$n\;:\left( {n - m + 1} \right)$

If $a,b,c$ are in H.P., then

$\frac{{b - c}}{{c - a}} = \frac{b}{a}$

$\frac{{c - a}}{{a - b}} = \frac{c}{b}$

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JEE • MATHEMATICS • SEQUENCE OF SERIES

Sequence of series practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 781-810 of 998 questions

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781 Easy

781 of 998

Application +4 | -1

The point on the graph

y=\log _2 \log _6\left\{2^{\sqrt{(2 x+1)}}+4\right\},

whose

y

-coordinate is 1 is

View answer and explanation

Correct option: D

(12,1)

Option D ((12,1)) is correct because substituting y=1 into the equation and solving for x yields x=12.

782 Easy

782 of 998

Application +4 | -1

The sum of the series

{1^2} + 1 + {2^2} + 2 + {3^2} + 3 + \ldots + {n^2} + n

View answer and explanation

Correct option: C

\frac{{n\left( {n + 1} \right)\left( {n + 2} \right)}}{3}

Option C is correct because the sum of the series can be expressed as the sum of squares and the sum of natural numbers, which simplifies to the given expression.

783 Easy

783 of 998

Understanding +4 | -1

S=\sum\limits_{{n=0}}^{\infty} \frac{(\log x)^{2 n}}{(2 n) !}

, then

S=

View answer and explanation

Correct option: C

\frac{1}{2}\left(x+x^{-1}\right)

Option C (

\frac{{1}}{{2}}\left(x+x^{{-1}}\right)

) is correct because the given series is the expansion of cosh(log x), which simplifies to the given expression.

784 Medium

784 of 998

Analyzing +4 | -1

p^{\text {th }}, q^{\text {th }}, r^{\text {th }}

, terms of a G.P. are the positive numbers

a, b, c

respectively then angle between the vectors

\log a^3 \hat{i}+\log b^3 \hat{j}+\log c^3 \hat{k}

and

(q-r) \hat{i}+(r-p) \hat{j}+(p-q) \hat{k}

is:

View answer and explanation

Correct option: A

\frac{\pi}{2}

Option A (

\frac{\pi}{2}

) is the correct answer because the dot product of the two vectors is zero, indicating they are orthogonal.

785 Medium

785 of 998

Application +4 | -1

{2^{\frac{3}{{{{\log }_3}x}}}} = \frac{1}{{64}},

then

x =

View answer and explanation

Correct option: C

\frac{1}{{\sqrt 3 }}

Option C (

\frac{1}{{\sqrt 3 }}

) is correct because by simplifying the exponential equation using logarithm properties, the value of

x

is found to be

\frac{1}{{\sqrt 3 }}

786 Easy

786 of 998

Application +4 | -1

The minimum value of

9\,\,{\tan ^2}\theta \,\, + \,\,4\,\,{\cot ^2}\theta

View answer and explanation

Correct option: D

\mathrm{12}

Option D is correct because the minimum value is found using the AM-GM inequality.

787 Easy

787 of 998

Application +4 | -1

The sum of

n

terms of two arithmetic series are in the ratio

2n + 3:6n + 5

, then the ratio of their 13th terms is

View answer and explanation

Correct option: A

53:155

Option A is correct because by setting up the ratio of the sums of the two arithmetic series and substituting n = 25, the ratio of their 13th terms is found to be 53:155.

788 Easy

788 of 998

Application +4 | -1

If the sum of first

n

terms of an A.P. is

\mathrm{cn}^2

, then the sum of squares of these

n

terms, is

View answer and explanation

Correct option: C

\frac{n\left(4 n^2-1\right)}{3} c^2

Option C (

\frac{n(4n^2-1)}{3}c^2

) is correct because it is the result of correctly summing the squares of the terms of the A.P.

789 Easy

789 of 998

Application +4 | -1

The sum of the series

1+3 x+6 x^2+10 x^3+\ldots \infty

, is

View answer and explanation

Correct option: D

\frac{1}{(1-x)^3}

Option D (

\frac{1}{(1-x)^3}

) is correct because it represents the sum of the series

1+3 x+6 x^2+10 x^3+\ldots \infty

, which is derived by successive differences of terms.

790 Easy

790 of 998

Application +4 | -1

If the sum of 12th and 22nd terms of an AP is 100, then the sum of the first 33 terms of the AP is

View answer and explanation

Correct option: B

1650

Option B (

1650

) is correct because the sum of the first 33 terms is calculated using the given information about the 12th and 22nd terms of the AP.

791 Medium

791 of 998

Application +4 | -1

Let

a=\log 25

and

b=\log 225

, then

\log \left(\frac{1}{81}\right)+\log \left(\frac{1}{2250}\right)

is equal to:

View answer and explanation

Correct option: C

2 a-3 b-1

Option C (

2a-3b-1

) is correct because by expressing the given logarithms in terms of a and b and simplifying, the expression

\log (\frac{{1}}{{81}})+\log (\frac{{1}}{{2250}})

simplifies to

2a-3b-1

792 Easy

792 of 998

Application +4 | -1

7th term of an AP is 40. Then, the sum of first 13 terms is

View answer and explanation

Correct option: A

520

Option A (520) is correct because the sum of an AP is equal to the number of terms multiplied by the middle term, which is given.

793 Easy

793 of 998

Application +4 | -1

The value of

{0.2^{{{\log }_{\sqrt 5 }}\left( {\frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \ldots } \right)}}

, is

View answer and explanation

Correct option: A

Option A (4) is correct because after simplifying the infinite geometric series and applying logarithm properties, the expression evaluates to 4.

794 Easy

794 of 998

Application +4 | -1

Let

T_r

be the

r^{\text {th }}

term of an

A . P

, for

r=1,2, \ldots

If for some positive integers

m, n

we have

T_m=1 / n

and

T_n=1 / m

, then

T_{m n}

equals

View answer and explanation

Correct option: C

Option C (1) is correct because it finds the value of T_mn using the properties of arithmetic progressions and the given conditions.

795 Medium

795 of 998

Application +4 | -1

The five geometric means between 576 and 9 are

View answer and explanation

Correct option: B

288,144,72,36,18

Option B (288, 144, 72, 36, 18) is correct because these are the five geometric means between 576 and 9, calculated using the common ratio of 1/2.

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796 Medium

796 of 998

Application +4 | -1

\sum\limits_{{n=1}}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}

, then

\sum\limits_{{n=1}}^{\infty} \frac{1}{(2 n-1)^2}

View answer and explanation

Correct option: A

\frac{\pi^2}{8}

Option A (

\frac{\pi^2}{8}

) is correct because it is derived by subtracting the sum of the reciprocals of even squares from the sum of the reciprocals of all squares.

797 Easy

797 of 998

Understanding +4 | -1

\frac{1}{x+1}+\frac{1}{2(x+1)^2}+\frac{1}{3(x+1)^3}+\ldots \infty=

View answer and explanation

Correct option: A

\log _v\left(1+\frac{1}{x}\right)

Option A (

\log _v\left(1+\frac{{1}}{{x}}\right)

) is correct because the given series is a logarithmic series expansion, and simplifying it results in the expression

\log _e\left(1+\frac{{1}}{{x}}\right)

798 Easy

798 of 998

Application +4 | -1

If the sum of three numbers in

A P

is -3 and their product is 8 , then the numbers are

View answer and explanation

Correct option: B

2,-1,-4

Option B (

2,-1,-4

) is correct because these numbers form an AP with a sum of -3 and a product of 8.

799 Easy

799 of 998

Analyzing +4 | -1

The sum of the series

\sum \limits_{n = 1}^\infty \frac{{2\;n}}{{\left( {2n + 1} \right)\;!}},

View answer and explanation

Correct option: B

{e^{ - 1}}

Option B (

{e^{ - 1}}

) is correct because the sum of the series can be expressed as the expansion of

e^{-1}

800 Easy

800 of 998

Application +4 | -1

Sum of infinite number of terms of a G.P. is 20 and sum of their squares is 100. The common ration of the G.P. is

View answer and explanation

Correct option: B

3/5

Option B (

3/5

) is correct because it is the common ratio that satisfies both the sum of infinite terms and the sum of their squares.

801 Medium

801 of 998

Application +4 | -1

\frac{{x - y}}{x} + \frac{1}{2}{\left( {\frac{{x - y}}{x}} \right)^2} + \frac{1}{3}{\left( {\frac{{x - y}}{x}} \right)^3} + ...

is equal to

View answer and explanation

Correct option: C

{\log _e}\left( {\frac{x}{y}} \right)\;

Option C (

{\log _e}\left( {\frac{x}{y}} \right)\;

) is correct because the given series is the expansion of

-\log_e(1 - \frac{{x-y}}{x})

which simplifies to

{\log _e}\left( {\frac{x}{y}} \right)

802 Easy

802 of 998

Application +4 | -1

a_{1}, a_{2}, a_{3}, \ldots . a_{n}

are in A.P. and

a_{1}+a_{4}+a_{7}+\ldots+a_{16}=114

, then

a_{1}+a_{6}+a_{11}+a_{16}

is equal to:

View answer and explanation

Correct option: B

Option B (76) is correct because the sum a_1 + a_6 + a_11 + a_16 can be expressed in terms of a_1 + a_16, which is derived from the given equation.

803 Easy

803 of 998

Application +4 | -1

\frac{{{e^x}}}{{1 - x}} = {B_0} + {B_1}x + {B_2}{x^2} + \ldots + {B_n}{x^n} + \ldots ,

then

{B_n} - {B_{n - 1}}

equals

View answer and explanation

Correct option: A

\frac{1}{{n\;!}}

Option A (

\frac{1}{{n\;!}}

) is correct because it is the coefficient of x^n in the expansion of e^x / (1-x).

804 Easy

804 of 998

Understanding +4 | -1

The product of

n

positive integers is 1 , then their sum is a positive integer, that is

View answer and explanation

Correct option: D

Never less than

n

Option A (

r

!) is correct because the product of

r

consecutive positive integers is always divisible by

r

factorial.

805 Easy

805 of 998

Application +4 | -1

The sum of the integers from 1 to 100 which are divisible by 3 and 5, is

View answer and explanation

Correct option: C

315

Option C (315) is correct because the sum of integers from 1 to 100 divisible by both 3 and 5 (i.e., divisible by 15) is 315.

806 Easy

806 of 998

Application +4 | -1

G_1

and

G_2

are two geometric means and

A

is the arithmetic mean inserted between two positive numbers then the value of

\frac{G_1^2}{G_2}+\frac{G_2^2}{G_1}

View answer and explanation

Correct option: C

2 A

Option C (

2A

) is correct because the expression simplifies to

a+b

, which is equal to

2A

where

A

is the arithmetic mean of

a

and

b

807 Easy

807 of 998

Application +4 | -1

{\log _3}\left\{ {{{\log }_6}\left( {\frac{{{x^2} + x}}{{x - 1}}} \right)} \right\} = 0

then

x =

View answer and explanation

Correct option: C

Option C (3) is correct because it is a solution to the equation obtained after simplifying the logarithmic expression.

808 Medium

808 of 998

Application +4 | -1

Given that

n

AM’s are inserted between two sets of numbers

a,\;2b\;\mathrm{and}\;2a,\;b\;\mathrm{where}\;a,\;b \in R.

. Suppose further that

m

th mean between these sets of numbers is same, then the ratio

a\;:b

equals

View answer and explanation

Correct option: D

m\;:\left( {n - m + 1} \right)

Option D (

m : (n - m + 1)

) is correct because it represents the ratio

a:b

derived from equating the

m

th means of the two sets of numbers.

809 Easy

809 of 998

Application +4 | -1

a,b,c

are in H.P., then

View answer and explanation

Correct option: A

\frac{{a - b}}{{b - c}} = \frac{a}{c}

Option A (

\frac{{a - b}}{{b - c}} = \frac{a}{c}

) is correct because it directly follows from the definition of Harmonic Progression.

810 Medium

810 of 998

Application +4 | -1

The next term of the sequence

1,3,6,10, \ldots

View answer and explanation

Correct option: C

Option C (15) is correct because the sequence follows a pattern where each term is the sum of the previous term and the next consecutive integer.

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