If $\left| a \right| < 1,\;b = \mathop \sum \limits_{k = 1}^\infty \frac{{{a^k}}}{k}$, then $a$ is equal to

$\mathop \sum \limits_{k = 1}^\infty \frac{{{{\left( { - 1} \right)}^{k - 1}}{b^k}}}{{k!}}$

$\mathop \sum \limits_{k = 1}^\infty \frac{{{{\left( { - 1} \right)}^k}{b^k}}}{k}$

$\mathop \sum \limits_{k = 1}^\infty \frac{{{{\left( { - 1} \right)}^k}{b^k}}}{{\left( {k - 1} \right)!}}$

$\mathop \sum \limits_{k = 1}^\infty \frac{{{{\left( { - 1} \right)}^{k - 1}}{b^k}}}{{\left( {k + 1} \right)!}}$

If $a,b,c$ are in A.P. and ${a^2},{b^2},{c^2}$ are in H.P., then

${b^2} = \sqrt {\left( {ac/8} \right)} $

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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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Showing 931-960 of 998 questions

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931 Easy

931 of 998

Application +4 | -1

\log _5 2, \log _5\left(2^x-3\right)

and

\log _5\left(\frac{17}{2}+2^{x-1}\right)

are in

A P

, then the value of

x

View answer and explanation

Correct option: C

Option C (3) is correct because it is the solution obtained by setting up and solving the equation derived from the arithmetic progression property.

932 Easy

932 of 998

Application +4 | -1

a,b,c,d,e,f

are A.M.’s between 2 and 12, then

a + b + c + d + e + f

is equal to

View answer and explanation

Correct option: B

Option B (42) is correct because the sum of the arithmetic means is calculated using the average of the first and last term.

933 Easy

933 of 998

Application +4 | -1

Let

a_1, a_2, a_3, \ldots, a_n \ldots

be an A.P. whose common difference is an integer and

S_{n}

denote the sum of its

n

terms. If

a_1=1, a_n=300

and

15 \leq n \leq 50

, then the ordered pair

\left(S_{n-4}, a_{n-4}\right)

is equal to

View answer and explanation

Correct option: A

(2490,248)

Option A (2490, 248) is correct because it is derived from finding the common difference and then calculating

S_{n-4}

and

a_{n-4}

934 Easy

934 of 998

Analyzing +4 | -1

The sum of the series

\sum\limits_{n = 1}^\infty {\frac{{{n^2} + 6n + 10}}{{\left( {2n + 1} \right)!}}}

is equal to

View answer and explanation

Correct option: B

\frac{{41}}{8}e - \frac{{19}}{8}{e^{ - 1}} - 10

Option B (

\frac{{41}}{8}e - \frac{{19}}{8}{e^{ - 1}} - 10

) is correct because the series can be decomposed and expressed in terms of exponential series expansions.

935 Easy

935 of 998

Application +4 | -1

The number which should be added to the numbers 2, 14, 62, so that the resulting numbers may be in GP, is

View answer and explanation

Correct option: B

Option B (2) is correct because adding 2 to the numbers 2, 14, and 62 results in a geometric progression (4, 16, 64).

936 Easy

936 of 998

Application +4 | -1

Three numbers are in GP such that their sum is 38 and their product is 1728. The greatest number among them is

View answer and explanation

Correct option: A

Option A (18) is correct because it is the largest number in the geometric progression that satisfies the given sum and product conditions.

937 Easy

937 of 998

Application +4 | -1

If the sum of the first ten terms of the series

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

\frac{16}{5} \mathrm{~m}

, then

m

equal to

View answer and explanation

Correct option: D

101

Option D (101) is correct because the sum of the given series can be simplified and calculated using the formula for the sum of squares of natural numbers.

938 Easy

938 of 998

Analyzing +4 | -1

If the non-zero numbers

x, y, z

are in

A P

, and

\tan ^{-1} x

\tan ^{-1} y, \tan ^{-1} z

are also in AP., then

View answer and explanation

Correct option: A

x=y=z

Option A (

x=y=z

) is correct because if

x, y, z

are in both A.P. and G.P., then they must be equal.

939 Easy

939 of 998

Application +4 | -1

\log \left( {x + y} \right) = \log 2 + \frac{1}{2}\log x + \frac{1}{2}\log y,\;\mathrm{then}

View answer and explanation

Correct option: B

x - y = 0

Option B (

x - y = 0

) is correct because it is the simplified form of the given logarithmic equation, indicating that x and y are equal.

940 Easy

940 of 998

Analyzing +4 | -1

Let the harmonic mean of two positive real numbers

a

and

b

be 4. If

q

is a positive real number such that

a, 5, q, b

is an arithmetic progression, then the value(s) of

|q-a|

is (are)

View answer and explanation

Correct option: B

2, 5

Option B (2, 5) is correct because solving the harmonic mean and arithmetic progression equations yields two possible values for |q-a|: 2 and 5.

941 Easy

941 of 998

Analyzing +4 | -1

The number 111…1 (91 times) is a/an

View answer and explanation

Correct option: C

Not prime

Option C (Not prime) is correct because the number 111…1 (91 times) can be factored into a product of two integers, thus it's not a prime number.

942 Easy

942 of 998

Analyzing +4 | -1

The sum of the series

a-(a+d)+(a+2 d)-(a+3 d)+\ldots

upto

(2 n+1)

terms, is

View answer and explanation

Correct option: C

a+n d

Option C (

a+n d

) is correct because the sum simplifies to this expression after pairing terms and canceling out the common difference.

943 Easy

943 of 998

Analyzing +4 | -1

Let

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

and

{T_n} = 1 + \left( {\frac{{q + 1}}{2}} \right) + {\left( {\frac{{q + 1}}{2}} \right)^2} + ... + {\left( {\frac{{q + 1}}{2}} \right)^n}

, where

q

is a real number and

q \ne 1.

{}^{101}{C_1} + {}^{101}{C_2} \cdot {S_1} + ... + {}^{101}{C_{101}} \cdot {S_{100}} = \alpha {T_{100}}

then

\alpha

is equal to

View answer and explanation

Correct option: A

{2^{100}}

Option A (

{2^{100}}

) is correct because it is the result of correctly simplifying the given expression using the properties of geometric series and binomial coefficients.

944 Easy

944 of 998

Application +4 | -1

{a^{1/x}} = {b^{1/y}} = {c^{1/z}}

and

a,b,c

are in geometrical progression, then

x,y,z

are in

View answer and explanation

Correct option: A

Option A is correct because if

{a^{1/x}} = {b^{1/y}} = {c^{1/z}}

and

a,b,c

are in GP, then

x, y, z

must be in AP.

945 Easy

945 of 998

Application +4 | -1

The sum of

{15^2} + {16^2} + {17^2} + ... + {30^2}

is equal to

View answer and explanation

Correct option: B

8440

Option B (8440) is correct because it is the result of calculating the sum of squares from 15 to 30 by subtracting the sum of squares from 1 to 14 from the sum of squares from 1 to 30.

Explore Other Chapters in MATHEMATICS

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DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs

946 Easy

946 of 998

Application +4 | -1

The sum of the numbers between 100 and 1000 which is divisible by 9 is

View answer and explanation

Correct option: A

55350

Option A (55350) is correct because the sum of the arithmetic progression of numbers divisible by 9 between 100 and 1000 is 55350.

947 Easy

947 of 998

Analyzing +4 | -1

|a|<1

and

|b|<1

, then the sum of the series

1+(1+a) b+\left(1+a+a^2\right) b^2+\left(1+a+a^2+a^3\right) b^3+\ldots

, is

View answer and explanation

Correct option: C

\frac{1}{(1-b)(1-a b)}

Option C (

\frac{{1}}{{(1-b)(1-ab)}}

) is correct because the sum of the series can be simplified to this expression using the formula for the sum of an infinite geometric series.

948 Easy

948 of 998

Application +4 | -1

An AP consists of 23 terms. If the sum of the three terms in the middle is 141 and the sum of the last three terms is 261, then the first term is

View answer and explanation

Correct option: D

Option D is correct because by setting up equations based on the given sums of terms in the AP and solving the system of equations, the first term can be determined to be 3.

949 Easy

949 of 998

Analyzing +4 | -1

Let

S=3+4+8+9+13+14+18+\ldots \ldots 40

terms. If

S=(102) m

, then

m

is equal to

View answer and explanation

Correct option: A

Option A (20) is correct because calculating the sum of the given series to 40 terms results in 2040, which equals (102)m, thus m = 20.

950 Easy

950 of 998

Analyzing +4 | -1

x = {\log _2}3

and

y = {\log _{1/2}}5,

then

View answer and explanation

Correct option: A

x > y

Option A (

x > y

) is correct because

x

is positive and

y

is negative, since the base of the logarithm in

y

is less than 1.

951 Easy

951 of 998

Application +4 | -1

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

are in AP, where

a_{i} > 0

for all

i

, then the value of

\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\cdots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}=

View answer and explanation

Correct option: A

\frac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n}}}

Option A (

\frac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n}}}

) is correct because rationalizing the terms and using the properties of AP simplifies the expression to

\frac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n}}}

952 Easy

952 of 998

Application +4 | -1

{A_1},{A_2}

are two A ∙ M’s between

a

and

b

and

{G_1},{G_2}

are two GM’s between

a

and

b

then

\frac{{{A_1} + {A_2}}}{{{G_1}{G_2}}}

View answer and explanation

Correct option: A

\frac{{a + b}}{{ab}}

Option A (

\frac{{a + b}}{{ab}}

) is correct because it is derived from the properties of Arithmetic and Geometric Progressions.

953 Easy

953 of 998

Application +4 | -1

H

be the H.M. between

a

and

b

, then the value of

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

View answer and explanation

Correct option: A

Option A (2) is correct because if

H

is the H.M. between

a

and

b

, then

\frac{H}{a} + \frac{H}{b} = 2

954 Easy

954 of 998

Application +4 | -1

Geometric mean of

7,{7^2},{7^3}, \ldots {7^n}

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the geometric mean of the given series is calculated by taking the nth root of the product of the terms.

955 Easy

955 of 998

Analyzing +4 | -1

\left| a \right| < 1,\;b = \sum \limits_{k = 1}^\infty \frac{{{a^k}}}{k}

, then

a

is equal to

View answer and explanation

Correct option: B

\sum \limits_{k = 1}^\infty \frac{{{{\left( { - 1} \right)}^{k - 1}}{b^k}}}{{k!}}

Option B (

\sum \limits_{k = 1}^\infty \frac{{{{\left( { - 1} \right)}^{k - 1}}{b^k}}}{{k!}}

) is correct because it represents the Maclaurin series expansion of

a

in terms of

b

, derived from

b = -\log(1-a)

956 Easy

956 of 998

Application +4 | -1

Let

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

be an A.P, such that

\frac{a_{1}+a_{2}+\ldots+a_{p}}{a_{1}+a_{2}+a_{3}+\ldots+a_{q}}=\frac{p^{3}}{q^{3}} ; p \neq q

. Then

\frac{a_{6}}{a_{21}}

is equal to:

View answer and explanation

Correct option: B

\frac{31}{121}

Option B (31/121) is correct because the ratio a6/a21 is calculated by using the given ratio of sums and properties of arithmetic progressions.

957 Easy

957 of 998

Application +4 | -1

If the 7th term of HP is

\frac{1}{{10}}

and the 12th term is

\frac{1}{{25}}

, then the 20th term is

View answer and explanation

Correct option: C

\frac{1}{{49}}

Option C (

\frac{1}{{49}}

) is correct because the 20th term of the Harmonic Progression is found by first finding the corresponding Arithmetic Progression and then inverting the result.

958 Easy

958 of 998

Analyzing +4 | -1

3 \leq a \leq 2015,3 \leq b \leq 2015

such that

\log _a b+6 \log _b a=5

, the number of ordered pairs

(a, b)

of integers is

View answer and explanation

Correct option: C

52

Option C (52) is correct because solving the logarithmic equation leads to two possible relationships between a and b, resulting in 52 ordered pairs.

959 Easy

959 of 998

Analyzing +4 | -1

a,b,c

are in A.P. and

{a^2},{b^2},{c^2}

are in H.P., then

View answer and explanation

Correct option: A

a = b = c

Option A (

a = b = c

) is correct because if a, b, c are in AP and a², b², c² are in HP, then the only possibility is that all three are equal.

960 Easy

960 of 998

Application +4 | -1

{a_1} = {a^2} = 2,\;{a_n} = {a_{n - 1}} - 1\left( {n > 2} \right),\;\mathrm{then}\;{a_5}

View answer and explanation

Correct option: B

- 1

Option B (-1) is correct because, given the recursive formula,

a_5

is calculated to be -1.

Revision notes and key takeaways

Review SEQUENCE OF SERIES in your textbook or class notes, then use this set to check your understanding.

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