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JEE • MATHEMATICS • SEQUENCE OF SERIES

Sequence of series practice · Easy

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Practice format and chapter quiz

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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751 Easy

751 of 998

Analyzing +4 | -1

x^2+9 y^2+25 z^2=x y z\left(\frac{15}{x}+\frac{5}{y}+\frac{3}{z}\right)

, then

x, y

and

z

are in

View answer and explanation

Correct option: D

H.P.

Option D (H.P.) is correct because the given equation implies x = 3y = 5z, which means x, y, and z are in H.P.

752 Easy

752 of 998

Application +4 | -1

The sum of 20 terms of a series of which every even term is 2 times the term before it, and every odd term is 3 times the term before it, the first term being unity is

View answer and explanation

Correct option: C

\left(\frac{3}{5}\right)\left(6^{10}-1\right)

Option C (

\left(\frac{{3}}{{5}}\right)\left(6^{10}-1\right)

) is correct because the series can be split into two geometric progressions, and their sums can be calculated.

753 Easy

753 of 998

Application +4 | -1

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

View answer and explanation

Correct option: A

3+2 \sqrt{2}

Option A (

3+2 \sqrt{{2}}

) is correct because by tripling the middle term and setting up an A.P., we can solve for the common ratio of the G.P.

754 Easy

754 of 998

Application +4 | -1

In a G.P. with alternatively positive and negative terms, any term is the A.M. of the next two terms. Then, the common ratio of the G.P. is

View answer and explanation

Correct option: C

- 2

Option C (-2) is correct because setting up the A.M. equation and solving the quadratic results in a common ratio of -2.

755 Easy

755 of 998

Understanding +4 | -1

{S_n} = {1^3} + {2^3} + ... + {n^3}

and

{T_n} = 1 + 2 + ... + n

, then

View answer and explanation

Correct option: C

{S_n} = T_n^2

Option C is correct because the sum of cubes of first n natural numbers is equal to the square of the sum of first n natural numbers.

756 Easy

756 of 998

Application +4 | -1

H.M. between

\frac{1}{28}

and

\frac{1}{10}

View answer and explanation

Correct option: A

\frac{1}{19}

Option A is correct because the harmonic mean between two numbers is calculated using the formula 2ab/(a+b).

757 Easy

757 of 998

Analyzing +4 | -1

n^{\text {th }}

term of the series

\frac{1^{2}}{2}+\frac{1^{2}+2^{2}}{4}+\frac{1^{2}+2^{2}+3^{2}}{6}+\ldots \ldots

View answer and explanation

Correct option: C

\frac{(n+1)(2 n+1)}{12}

Option C ((n+1)(2n+1)/12) is correct because it represents the nth term of the series, derived by summing the squares of the first n natural numbers and dividing by 2n.

758 Easy

758 of 998

Application +4 | -1

The value of

{a^{\frac{{{{\log }_b}\left( {{{\log }_b}x} \right)}}{{{{\log }_b}a}}}},

View answer and explanation

Correct option: B

{\log _b}x

Option B (

{\log _b}x

) is correct because the expression simplifies using properties of logarithms and exponents.

759 Easy

759 of 998

Application +4 | -1

The sum of a GP with common ratio 3 is 364 and last term is 243, then the number of terms is

View answer and explanation

Correct option: A

Option A (6) is correct because it is the number of terms that satisfies the given conditions for the geometric progression.

760 Easy

760 of 998

Application +4 | -1

If 2/3,

k

, 5/8 are in AP, then value of

k

View answer and explanation

Correct option: D

31/48

Option D (

31/48

) is correct because in an arithmetic progression, the middle term is the average of the first and third terms.

761 Easy

761 of 998

Application +4 | -1

Which of the following sequences is an A.P. with common difference 3?

View answer and explanation

Correct option: B

a_n=3 n+5

Option B (

a_n=3n+5

) is correct because it represents a linear expression in 'n' with a coefficient of 3, indicating an arithmetic progression with a common difference of 3.

762 Easy

762 of 998

Understanding +4 | -1

\sum \limits_{n = 0}^\infty \frac{{{{\left( {{{\log }_e}x} \right)}^n}}}{{n\;!}}\;\mathrm{is\;equal\;to}

View answer and explanation

Correct option: B

x

Option B (

x

) is correct because the given series is the Taylor series expansion of

e^y

where

y = \log_e x

763 Easy

763 of 998

Analyzing +4 | -1

{a_1},{a_2}, \ldots {a_n}

are in arithmetic progression, where

{a_1} > 0

for all

i

.

Then,

\frac{1}{{\sqrt {{a_1}} + \sqrt {{a_2}} }} + \frac{1}{{\sqrt {{a_2}} + \sqrt {{a_3}} }} + ... + \frac{1}{{\sqrt {{a_{n - 1}}} + \sqrt {{a_n}} }}

is equal to

View answer and explanation

Correct option: B

\frac{{n - 1}}{{\sqrt {{a_1}} + \sqrt {{a_n}} }}

Option B (

\frac{{n - 1}}{{\sqrt {{{{a_1}}}} + \sqrt {{{{a_n}}}} }}

) is the correct answer. The sum telescopes after rationalizing each term, and the common difference allows simplification to the final expression.

764 Easy

764 of 998

Analyzing +4 | -1

Suppose four distinct positive numbers

a_1, a_2, a_3, a_4

are
in GP. Let

b_1=a_1, b_2=b_1+a_2, b_3=b_2+a_3

and

b_4=b_3+a_4

Statement- 1 The numbers

b_1, b_2, b_3, b_4

are neither in AP nor in GP.

Statement -2 The numbers

b_1, b_2, b_3, b_4

are in

\mathrm{HP}

View answer and explanation

Correct option: C

Statement- 1 is true, Statement-2 is false

Option C (Statement-1 is true, Statement-2 is false) is correct because the numbers

b_1, b_2, b_3, b_4

are neither in AP nor in GP nor in HP.

765 Easy

765 of 998

Application +4 | -1

If the ratio of

n^{\text {th }}

terms of two A.P.S is

(2 n+8):(5 n-3)

, then the ratio of the sums of their

n

terms is

View answer and explanation

Correct option: C

(2 n+18):(5 n-1)

Option C (

(2n+18):(5n-1)

) is correct because the ratio of the sums of their n terms is obtained by replacing n by (n+1)/2 in the given ratio.

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766 Easy

766 of 998

Application +4 | -1

The G.M. of the numbers

3,3^{2}, 3^{3} \ldots \ldots .3^{n}

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the geometric mean of a geometric progression is calculated by raising the product of the terms to the power of 1/n.

767 Easy

767 of 998

Application +4 | -1

If the sum of three numbers in A.P. is 24 and their product is 440, then common difference of the A.P. can be

View answer and explanation

Correct option: A

Option A (3) is correct because by setting up equations based on the given sum and product, we can solve for the common difference.

768 Easy

768 of 998

Application +4 | -1

{x^{{{\log }_x}\left( {{x^2} - 4x + 5} \right)}} = \left( {x - 1} \right),

then

x =

View answer and explanation

Correct option: B

Option B (2) is correct because substituting x=2 into the equation satisfies the given condition.

769 Easy

769 of 998

Application +4 | -1

Let

{S_n}

denote the sum of the first

n

terms of an AP. If

{S_4} = 16

and

{S_6} = - \,48

then

{S_{10}}

is equal to

View answer and explanation

Correct option: C

-320

Option C (-320) is correct because it is the sum of the first 10 terms of the arithmetic progression, calculated using the given sums of the first 4 and 6 terms.

770 Easy

770 of 998

Application +4 | -1

2{\log _8}a = x,{\log _2}2a = y

and

y - x = 4,

then

x =

View answer and explanation

Correct option: D

Option D (6) is correct because solving the system of logarithmic equations leads to x = 6.

771 Easy

771 of 998

Application +4 | -1

The sum of first two terms of an infinite G.P. is 1 and every term is twice the sum of the successive terms. Its first term is

View answer and explanation

Correct option: C

3/4

Option C

3/4

is correct because it is found by setting up equations based on the given information and solving for the first term.

772 Easy

772 of 998

Application +4 | -1

{\log _{30}}3 = x,{\log _{30}}5 = y,

then

{\log _{30}}8 =

View answer and explanation

Correct option: A

3\left( {1 - x - y} \right)

Option A,

3(1 - x - y)

, is correct because it is derived by expressing

{\log_{30}}8

in terms of

{\log_{30}}3

and

{\log_{30}}5

773 Easy

773 of 998

Application +4 | -1

If an

A P

is given by

3, a_2, \ldots, 20

, then which of the following is the sum of second and second last terms?

View answer and explanation

Correct option: C

Option C (23) is correct because the sum of the second and second-to-last terms in an AP is equal to the sum of the first and last terms.

774 Easy

774 of 998

Analyzing +4 | -1

Let

x > 1

and for each positive integer

n

, define

\begin{aligned} s_n(x)= & \frac{x}{x+1}+\frac{x^2}{(x+1)\left(x^2+1\right)}+\frac{x^4}{(x+1)\left(x^2+1\right)\left(x^4+1\right)}+ \\ & \cdots+\frac{x^{2-1}}{(x+1)\left(x^2+1\right) \cdots\left(x^{2^{6-1}}+1\right)} \end{aligned}

Then

\lim \limits_{{n \rightarrow \infty}} s_n(x)

is equal to

View answer and explanation

Correct option: A

Option A is correct. By manipulating the series and taking the limit as n approaches infinity, the sum converges to 1.

775 Easy

775 of 998

Application +4 | -1

If the interior angles of a polygon are in A.P. with common difference

5^\circ

and smallest angle is

120^\circ ,

then the number of sides of the polygon is

View answer and explanation

Correct option: B

Option B (9) is correct because solving the quadratic equation for the number of sides results in two possible values, but only one is valid given the constraint on the largest angle.

776 Easy

776 of 998

Application +4 | -1

x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} \cdot x^{\frac{1}{8}} \cdot x^{\frac{1}{16}} \ldots=

View answer and explanation

Correct option: C

x

Option C is correct. The expression

x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} \cdot x^{\frac{1}{8}} \cdot x^{\frac{1}{16}} \ldots

simplifies to

x

by summing the infinite geometric series in the exponent.

777 Easy

777 of 998

Application +4 | -1

\sum \limits_{k = 1}^5 \frac{{{1^3} + {2^3} + ... + {k^3}}}{{1 + 3 + 5 + ... + \left( {2k - 1} \right)}}

is equal to

View answer and explanation

Correct option: A

22.5

Option A (22.5) is correct because it is the result of simplifying the summation and calculating the sum of the series.

778 Easy

778 of 998

Application +4 | -1

x,\;y,\;z

are in HP, then the value of expression

\log \left( {x + z} \right) + \log \left( {x - 2y + z} \right)\;

will be

View answer and explanation

Correct option: B

2\log \left( {x - z} \right)

Option B (

2\log \left( {x - z} \right)

) is correct because the given expression simplifies to

2\log(x-z)

when x, y, and z are in Harmonic Progression (HP).

779 Easy

779 of 998

Application +4 | -1

If the sum of the series

20+19 \frac{3}{5}+19 \frac{1}{5}+18 \frac{4}{5}+\ldots .

. upto

n

terms is 488 and the

n^{\text {th }}

term is negative, then the value of

n^{\text {th }}

term is

View answer and explanation

Correct option: B

-4

The correct answer is -4 because it is the negative nth term of the series that satisfies the given sum of 488.

780 Easy

780 of 998

Analyzing +4 | -1

Sum of the first

n

terms of the series

\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots

is equal to

View answer and explanation

Correct option: C

n+2^{-n}-1

Option C (

n+2^{-n}-1

) is correct because the sum can be rewritten as a difference of 1 and a geometric progression, which simplifies to the given expression.

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