The sum of the series $2\left[ {{7^{ - 1}} + {3^{ - 1}}{{.7}^{ - 3}} + {5^{ - 1}}{{.7}^{ - 5}} + ...} \right]$ is

${\log _e}\left( {\frac{3}{4}} \right)$

$2{\log _e}\left( {\frac{3}{4}} \right)$

$2{\log _e}\left( {\frac{4}{3}} \right)$

The sum to $n$ terms of the series ${2^2} + {4^2} + {6^2} + ...$ is

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

$\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}$

$\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{9}$

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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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121 Medium

121 of 998

Application +4 | -1

The sum of the series

2\left[ {{7^{ - 1}} + {3^{ - 1}}{{.7}^{ - 3}} + {5^{ - 1}}{{.7}^{ - 5}} + ...} \right]

View answer and explanation

Correct option: A

{\log _e}\left( {\frac{4}{3}} \right)\;

Option A is correct because the series can be recognized as the expansion of a logarithmic function, specifically related to the inverse hyperbolic tangent.

122 Easy

122 of 998

Analyzing +4 | -1

Let

S = 2 + \frac{6}{7} + \frac{{12}}{{{7^2}}} + \frac{{20}}{{{7^3}}} + \frac{{30}}{{{7^4}}} + \ldots

Then

4 S

is equal to

View answer and explanation

Correct option: C

{\left( {\frac{7}{3}} \right)^3}

The correct answer is C (

{\left( {\frac{7}{3}} \right)^3}

) because it is the value of 4S calculated by manipulating the series and using the formula for the sum of an infinite geometric series.

123 Easy

123 of 998

Application +4 | -1

a, b, c

are in H.P. then

a-\frac{b}{2}, \frac{b}{2}, c-\frac{b}{2}

are in

View answer and explanation

Correct option: B

G.P.

Option B (G.P.) is correct because the product of the first and third terms equals the square of the second term, satisfying the condition for a geometric progression.

124 Medium

124 of 998

Understanding +4 | -1

Assertion: If

a, b, c

are in A.P., then

b+c, c+a, a+b

are in A.P.

Reason: If

a, b, c

are in A.P., then

10^{{a}}, 10^{{b}}, 10^{{c}}

are in GP.

View answer and explanation

Correct option: B

Assertion is correct, reason is correct; reason is not a correct explanation for assertion

Option B is correct because both the assertion and reason are correct mathematical statements, but the reason does not directly explain the assertion.

125 Easy

125 of 998

Application +4 | -1

If a is positive and if

A

and

G

are the arithmetic mean and the geometric mean of the roots of

{x^2} - 2ax + {a^2} = 0

respectively, then

View answer and explanation

Correct option: A

A = G

Option A (

A = G

) is correct because the arithmetic mean and geometric mean of the roots of the given equation are equal, as the roots are equal.

126 Easy

126 of 998

Application +4 | -1

Three numbers are in AP such that their sum is 18 and sum of their squares is 158. The greatest number among them is

View answer and explanation

Correct option: B

Option B (11) is correct because by setting up the equations based on the given conditions for an AP, we find the greatest number to be 11.

127 Easy

127 of 998

Application +4 | -1

\frac{1}{{{{\log }_x}10}} = \frac{2}{{{{\log }_a}10}} - 2,

then

x =

View answer and explanation

Correct option: D

\frac{{{a^2}}}{{100}}

Option D (

\frac{{{a^2}}}{{100}}

) is correct because by using the change of base formula and simplifying, we can solve for x and find its value to be a²/100.

128 Easy

128 of 998

Application +4 | -1

S_{n}=\sum_{r=0}^{n} \frac{1}{{ }^{n} C_{r}}

and

t_{n}=\sum_{r=0}^{n} \frac{r}{{ }^{n} C_{r}}

, then

\frac{t_{n}}{S_{n}}

is equal to

View answer and explanation

Correct option: D

\frac{1}{2} n

Option C (

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

) is correct because it correctly calculates the sum of the series when n is odd, using the given sum for when n is even.

129 Easy

129 of 998

Application +4 | -1

\frac{{3 + 5 + 7 + \ldots + n\;\mathrm{terms}}}{{5 + 8 + 11 + \ldots + 10\;\mathrm{terms}}} = 7,

then the value of

n

View answer and explanation

Correct option: A

Option A (35) is correct because solving the equation derived from the arithmetic progression sum formula yields n = 35.

130 Medium

130 of 998

Understanding +4 | -1

The value of the series

x\,\,{\log _e}a + \frac{{{x^3}}}{{3!}}\,{\left( {{{\log }_e}a} \right)^3} + \frac{{{x^5}}}{{5!}}\,\,{\left( {{{\log }_e}a} \right)^5} + ...

View answer and explanation

Correct option: C

\sinh \,\,\left( {x{{\log }_e}\,\,a} \right)

Option C (

\sinh \,\,\left( {x{{\log }_e}\,a} \right)

) is correct because the given series is the Taylor series expansion of sinh(x log a).

131 Hard

131 of 998

Application +4 | -1

x,y,z

are three consecutive positive integers, then

{\log _e}\sqrt x + {\log _e}\sqrt z + \left( {\frac{1}{{2xz + 1}}} \right) + \frac{1}{3}{\left( {\frac{1}{{2xz + 1}}} \right)^3} + \frac{1}{5}{\left( {\frac{1}{{2xz + 1}}} \right)^5} + \;...

View answer and explanation

Correct option: B

{\log _e}y

Option B (

{\log _e}y

) is correct because the expression simplifies to

{\log _e}\sqrt{xz} + \frac{1}{2}{\log _e}\left[ {\frac{{1 + \frac{1}{{2xz + 1}}}}{{1 - \frac{1}{{2xz + 1}}}}} \right]

which further simplifies to

{\log _e}y

when x, y, and z are consecutive integers.

132 Easy

132 of 998

Application +4 | -1

{S_n} = nP + \frac{1}{2}n\left( {n - 1} \right){\cal Q}

, where

{S_n}

denotes the sum of the first

n

terms of an AP, then the common difference is

View answer and explanation

Correct option: D

{\cal Q}

Option D (

{\cal Q}

) is correct because the common difference of an AP can be found by subtracting consecutive terms, which simplifies to

{\cal Q}

133 Easy

133 of 998

Application +4 | -1

If the sum of the series

2+\frac{5}{x}+\frac{25}{x^2}+\frac{125}{x^3}

+\ldots

is finite, then

View answer and explanation

Correct option: A

|x|>5

Option A (

|x|>5

) is correct because the given series is a geometric series, and for its sum to be finite, the absolute value of the common ratio must be less than 1.

134 Easy

134 of 998

Application +4 | -1

The sum of all the numbers of the form

n^3

which lie between 100 and 10000 , is

View answer and explanation

Correct option: B

53261

Option B (53261) is correct because the sum of the cubes of integers between the given range is calculated to be 53261.

135 Easy

135 of 998

Application +4 | -1

{\log _{10}}x = y,

then

{\log _{{{10}^3}}}{x^2}

equals

View answer and explanation

Correct option: B

\frac{2}{3}y

Option B (2/3 y) is correct because it is the simplified form of the logarithm expression using the properties of logarithms.

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136 Medium

136 of 998

Application +4 | -1

If the first, second and last terms of an AP are

a, b, c

respectively, then the sum is

View answer and explanation

Correct option: C

\frac{(a+c)(b+c-2 a)}{2(b-a)}

Option C (

\frac{(a+c)(b+c-2a)}{2(b-a)}

) is correct because it is the sum of the AP, derived using the given first, second, and last terms.

137 Medium

137 of 998

Application +4 | -1

If roots of the equation

{x^3} - 12\;{x^2} + 39\;x - 28 = 0

are in A.P., then their common difference will be

View answer and explanation

Correct option: C

\pm 3

Option C (

\pm 3

) is correct because substituting the roots in A.P. into the cubic equation allows us to solve for the common difference.

138 Medium

138 of 998

Application +4 | -1

a,b,c

be in G.P. and

a + x,b + x,c + x

in H.P., then the value of

x

is (

a,b,c

are distinct numbers)

View answer and explanation

Correct option: B

b

Option B (

b

) is correct because substituting

b^2 = ac

and simplifying the equation derived from the harmonic progression leads to

x = b

139 Easy

139 of 998

Application +4 | -1

Let

a_1, a_2, \ldots

be in AP and

q_1, q_2, \ldots

be in GP. If

a_1=q_1=2

and

a_{10}=q_{10}=3

, then

View answer and explanation

Correct option: C

a_7 q_{19}=a_{19} q_{10}

Option C (

a_7 q_{19}=a_{19} q_{10}

) is correct because by calculating the values of

a_7

a_{19}

q_7

q_{10}

, and

q_{19}

, we can verify that

a_7 q_{19}=a_{19} q_{10}

140 Medium

140 of 998

Application +4 | -1

{\log _{10}}5 = x,

then

{\log _5}1250

equals to

View answer and explanation

Correct option: C

3 + \frac{1}{x}

Option C is correct because by using the properties of logarithms and the given value of log₁₀5, we can express log₅1250 as 3 + 1/x.

141 Easy

141 of 998

Application +4 | -1

Sum of the series

S=1+\frac{1}{2}(1+2)+

\frac{1}{3}(1+2+3)+\frac{1}{4}(1+2+3+4)+\ldots

upto 20 terms is

View answer and explanation

Correct option: C

115

Option C (115) is correct because the sum of the series can be simplified by finding a general term and then summing the resulting arithmetic series.

142 Easy

142 of 998

Application +4 | -1

The sum to

n

terms of the series

\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+\ldots

View answer and explanation

Correct option: A

\frac{6 n}{n+1}

Option A (

\frac{{6 n}}{{n+1}}

) is correct because the series can be simplified to a telescoping series, and the sum to

n

terms is

\frac{{6n}}{{n+1}}

143 Easy

143 of 998

Application +4 | -1

\frac{{a + b}}{2},b,\frac{{b + c}}{2}

are in HP, then

a,b,c

are in

View answer and explanation

Correct option: C

Option C (GP) is correct because the given condition leads to b² = ac, which is the condition for a, b, and c to be in GP.

144 Easy

144 of 998

Application +4 | -1

a>0

, then

\sum\limits_{n=1}^{\infty}\left(\frac{a}{a+1}\right)^n

equals

View answer and explanation

Correct option: D

a

Option D (

a

) is correct because the given expression is an infinite geometric series whose sum can be directly calculated using the formula

\frac{a}{1-r}

145 Medium

145 of 998

Application +4 | -1

The unit digit of

17^{1983}+11^{1983}-7^{1983}

View answer and explanation

Correct option: A

Option A (1) is correct because by analyzing the unit digits of each term separately and then summing them, we find the unit digit of the entire expression to be 1.

146 Easy

146 of 998

Application +4 | -1

Given sum of the first

n

terms of an A.P. is

2 n+3 n^{2}

. Another

View answer and explanation

Correct option: B

6 n^{2}-n

Option B (6n² - n) is correct because it produces an A.P. where the first term and the difference between the second and first term align with the given series.

147 Easy

147 of 998

Application +4 | -1

The sum of 100 terms of the series

0.9 + 0.09 + 0.009 \ldots

will be

View answer and explanation

Correct option: A

1 - {\left( {\frac{1}{{10}}} \right)^{100}}

Option A (

1 - {\left( {\frac{1}{{10}}} \right)^{100}}

) is correct because it represents the sum of the given series, which is a geometric progression, up to 100 terms.

148 Easy

148 of 998

Application +4 | -1

The sum to

n

terms of the series

\frac{4}{3} + \frac{{10}}{9} + \frac{{28}}{{27}} + ...

View answer and explanation

Correct option: B

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

Option B is correct because the sum is calculated by separating the series into two parts: a constant series and a geometric series, and then summing each part.

149 Medium

149 of 998

Application +4 | -1

The sum to

n

terms of the series

{2^2} + {4^2} + {6^2} + ...

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the sum of the series

{2^2} + {4^2} + {6^2} + ...

n

terms is equal to

4

times the sum of the squares of the first

n

natural numbers.

150 Easy

150 of 998

Application +4 | -1

If the sum of the series

3+\frac{1}{4}(3+d)+\frac{1}{4^2}(3+2 d)+\ldots

\infty

is 8 , then the value of

d

, is

View answer and explanation

Correct option: A

Option A (9) is correct because it is the value of d that satisfies the given infinite series equation.

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