If $a,\;b,\;c,\;d$ be in HP, then

${a^2} + {c^2} > {b^2} + {d^2}$

${a^2} + {d^2} > {b^2} + {c^2}$

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

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

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

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

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JEE • MATHEMATICS • SEQUENCE OF SERIES

Sequence of series practice · Easy

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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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781 Easy

781 of 998

Application +4 | -1

x=2^{1 / 3}-2^{-1 / 3}

, then

2 x^{3}+6 x=

View answer and explanation

Correct option: C

Option C (3) is correct because cubing the expression for x and simplifying leads to

2x^3 + 6x = 3

782 Easy

782 of 998

Analyzing +4 | -1

a^{\log _b c}=3 \cdot 3^{\log _4 3} \cdot 3^{\log _4 3^{\log _4 3}} \cdot 3^{\log _4 3^{\log _4 3^{\log _4 3}}} ...\infty,

where

a, b, c \in Q,

the value of

a b c

View answer and explanation

Correct option: C

Option C (16) is correct because by simplifying the infinite series in the exponent and comparing the bases, we find a=3, b=4/3, and c=4, so abc=16.

783 Easy

783 of 998

Application +4 | -1

a G P

, if

\left\{a_n\right\}, a_1=3, a_n=96

and

S_n=189

, then the value of

r

View answer and explanation

Correct option: B

Option B (2) is correct because the common ratio 'r' is correctly calculated using the formula for the sum of a geometric progression.

784 Easy

784 of 998

Application +4 | -1

\left( {1 - p} \right)\left( {1 + 2x + 4{x^2} + 8{x^3} + 16{x^4} + 32{x^5}} \right) = 1 - {p^6},p \ne 1

then a value of

p/x

View answer and explanation

Correct option: B

Option B (2) is correct because by simplifying the given equation, we find that p = 2x, hence p/x = 2.

785 Easy

785 of 998

Analyzing +4 | -1

a,\;b,\;c,\;d

be in HP, then

View answer and explanation

Correct option: C

ac + bd > {b^2} + {c^2}

The correct answer is C (

ac + bd > {b^2} + {c^2}

) because it is derived from the properties of Harmonic Progression and the relationship between Arithmetic Mean, Geometric Mean, and Harmonic Mean.

786 Easy

786 of 998

Analyzing +4 | -1

I_n=\int_0^\pi \frac{1-\sin 2 n x}{1-\cos 2 x} d x

, then

I_1, I_2, I_3, \ldots

are in

View answer and explanation

Correct option: A

Option A (AP) is correct because the difference between consecutive terms

I_{n+1} - I_n

is constant, indicating an arithmetic progression.

787 Easy

787 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.

788 Easy

788 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.

789 Easy

789 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.

790 Easy

790 of 998

Application +4 | -1

The sequence

\left\{a_n\right\}

is a GP such that

\frac{a_4}{a_6}=\frac{1}{4} \quad \text { and } \quad a_2+a_5=216

. If the common ratio is positive, then the first term is

View answer and explanation

Correct option: A

Option A (12) is correct because solving the GP equations gives the first term as 12.

791 Easy

791 of 998

Analyzing +4 | -1

Let

S_n(1 \leq n \leq 9)

denotes the sum of

n

terms of the series

1+22+333+\ldots+\underbrace{999 \ldots 9}_{9 \text { times }}

, then for

2 \leq n \leq 9

View answer and explanation

Correct option: C

9\left(S_n-S_{n-1}\right)=n\left(10^n-1\right)

Option C (

9(S_n-S_{n-1})=n(10^n-1)

) is correct because the difference between consecutive sums simplifies to a geometric series which can be expressed in the given form.

792 Easy

792 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.

793 Easy

793 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.

794 Easy

794 of 998

Analyzing +4 | -1

a, a_1, a_2, \ldots, a_{2 n}, b

are in A.P. and

a, g_1, g_2, g_3, \ldots \ldots . g_{2 n}

b

are in G.P. and

h

is the H.M. of

a

and

b

, then

\frac{a_1+a_{2 n}}{g_1 g_{2 n}}+\frac{a_2+a_{2 n-1}}{g_2 g_{2 n-1}}+\ldots+\frac{a_n+a_{n+1}}{g_n \cdot g_{n+1}}

is equal to

View answer and explanation

Correct option: A

\frac{2 n}{h}

Option A (

\frac{2n}{h}

) is correct because the expression simplifies to this value using the properties of arithmetic and geometric progressions, and the harmonic mean.

795 Easy

795 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.

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796 Easy

796 of 998

Analyzing +4 | -1

a,b,c,d

are in G.P. and

{a^x} = {b^y} = {c^z} = {d^u},

then

x,y,z,u

are in

View answer and explanation

Correct option: C

H.P.

Option C (H.P.) is correct because if

a, b, c, d

are in G.P. and

{a^x} = {b^y} = {c^z} = {d^u}

, then

x, y, z, u

are in H.P.

797 Easy

797 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.

798 Easy

798 of 998

Analyzing +4 | -1

The

50^{\text {th }}

term of the serie's

2+3+6+11+18+\ldots .

. Is

View answer and explanation

Correct option: D

49^2+2

Option D (

49^2+2

) is correct because the 50th term of the series is

49^2 + 2

799 Easy

799 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.

800 Easy

800 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.

801 Easy

801 of 998

Analyzing +4 | -1

{\left( {2.3} \right)^x} = {\left( {0.23} \right)^y} = 1000,

then

\frac{1}{x} - \frac{1}{y}

equals to

View answer and explanation

Correct option: C

\frac{1}{3}

Option C (

\frac{1}{3}

) is correct because it is the result of solving the logarithmic equations derived from the given conditions.

802 Easy

802 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.

803 Easy

803 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.

804 Easy

804 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)

805 Easy

805 of 998

Analyzing +4 | -1

\frac{1}{{1!\left( {n - 1} \right)!}} + \frac{1}{{3!\left( {n - 3} \right)!}} + \frac{1}{{5!\left( {n - 5} \right)!}} + ...

is equal to

View answer and explanation

Correct option: C

\frac{{{2^{n - 1}}}}{{n!}}

for all values of

n

Option C (

\frac{{{2^{n - 1}}}}{{n!}}

for all values of

n

) is correct because the given expression can be simplified using binomial coefficients.

806 Easy

806 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.

807 Easy

807 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}

808 Easy

808 of 998

Analyzing +4 | -1

f(x+y)=f(x) f(y)

for all

x, y \in N

and

\sum\limits_{x=1}^{\infty} f(x)=2

, then the value of

\frac{f(4)}{f(2)}

View answer and explanation

Correct option: D

\frac{4}{9}

Option D (

\frac{4}{9}

) is correct because, using the given functional equation and the sum to infinity, we can determine

f(1)

and then calculate

\frac{f(4)}{f(2)}

809 Easy

809 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.

810 Easy

810 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.

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