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 · 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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811 Easy

811 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).

812 Easy

812 of 998

Analyzing +4 | -1

a, b, c, d

and

p

are distinct real numbers such that

\left(a^2+b^2+c^2\right) p^2-2 p(a b+b c+c d)+\left(b^2+c^2+d^2\right) \leq 0

then

a, b, c, d

are in

View answer and explanation

Correct option: B

G.P

The numbers

a, b, c, d

are in G.P. because the given inequality implies that

\frac{b}{a}=\frac{c}{b}=\frac{d}{c}=p

, which is the condition for a geometric progression.

813 Easy

813 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.

814 Easy

814 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.

815 Easy

815 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

816 Easy

816 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.

817 Easy

817 of 998

Analyzing +4 | -1

Let

a,\;p,\;q,\;r,\;s \in R\; \sim \;\left\{ 0 \right\}.

3{a^2} + 2\left( {\frac{1}{p} - \frac{1}{s}} \right)a + \frac{1}{{{p^2}}} + \frac{1}{{{q^2}}} + \frac{1}{{{r^2}}} - 2\left( {\frac{1}{{pq}} + \frac{1}{{qr}} + \frac{1}{{rs}}} \right) \le 0

for some real

a,

then

p,\;q,\;r,\;s

are in

View answer and explanation

Correct option: C

Option C (HP) is correct because the given inequality implies that the reciprocals of p, q, r, and s are in arithmetic progression, thus p, q, r, and s are in harmonic progression.

818 Easy

818 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.

819 Easy

819 of 998

Analyzing +4 | -1

The sum of the infinite series

1+\frac{5}{6}+\frac{12}{6^2}+\frac{22}{6^3}+\frac{35}{6^4}+\frac{51}{6^5}+\frac{70}{6^6}+\ldots

. is equal to

View answer and explanation

Correct option: C

\frac{288}{125}

Option A (

f

is not differentiable at

x=4

) is not true because after finding the value of b and checking differentiability, f is found to be differentiable at x=4.

820 Easy

820 of 998

Application +4 | -1

{x_1},\;{x_2},\;{x_3}

as well as

{y_1},\;{y_2},\;{y_3}

are in GP with the same common ratio, then the points

\left( {{x_1},\;{y_1}} \right),\;\left( {{x_2},\;{y_2}} \right)

and

\left( {{x_3},\;{y_3}} \right)

View answer and explanation

Correct option: A

Lie on a straight line

Option A (Lie on a straight line) is correct because the area of the triangle formed by the points is zero, indicating collinearity.

821 Easy

821 of 998

Application +4 | -1

S

denotes the sum to infinity and

{S_n}

the sum of

n

terms of the series

1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots ,

such that

S - {S_n} < \frac{1}{{1000}},

then the least value of

n

View answer and explanation

Correct option: D

Option D (11) is correct because the least value of n that satisfies the condition S - Sn < 1/1000 is 11.

822 Easy

822 of 998

Application +4 | -1

If the sequence

9,12,15,18, \ldots

is an AP, then its general term is

View answer and explanation

Correct option: D

3 n+6

Option D (

3n+6

) is correct because it represents the general term of the arithmetic progression with a first term of 9 and a common difference of 3.

823 Easy

823 of 998

Application +4 | -1

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24 , then the length of its smallest side is

View answer and explanation

Correct option: B

Option B (6) is correct because it is the length of the smallest side of the right-angled triangle, calculated by using the properties of arithmetic progression and the area of the triangle.

824 Easy

824 of 998

Analyzing +4 | -1

If positive numbers

{a^{ - 1}},\;{b^{ - 1}},\;{c^{ - 1}}

are in AP, then the product of roots of the equation

{x^2} - kx + 2{b^{101}} - {a^{101}} - {c^{101}} = 0

\left( {k \in R} \right)

View answer and explanation

Correct option: B

< 0

Option B (

<0

) is correct because the product of the roots is equal to

2b^{101} - a^{101} - c^{101}

, which is less than 0 since a, b, c are in HP.

825 Easy

825 of 998

Analyzing +4 | -1

Let

a, b, c

be three positive prime numbers. The progression in which

\sqrt{a}, \sqrt{b}, \sqrt{c}

can be three terms (not necessarily consecutive), is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because it can be shown that

\sqrt{a}, \sqrt{b}, \sqrt{c}

cannot be terms in AP, GP, or HP when a, b, and c are distinct positive prime numbers.

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826 Easy

826 of 998

Analyzing +4 | -1

Let

{a_1},\,\,{a_2},......{a_{21}}

be an AP such that

\sum\limits_{n = 1}^{20} {\frac{1}{{{a_n}{a_{n + 1}}}}} = \frac{4}{9}.

If the sum of this AP is 189, then

{a_6}{a_{16}}

is equal to

View answer and explanation

Correct option: B

Option B (72) is correct because solving the given equations for the arithmetic progression leads to a_6 * a_16 = 72.

827 Easy

827 of 998

Application +4 | -1

\left| x \right| < 1,\;

then the sum of the series

1 + 2x + 3{x^2} + 4{x^3} + ...\infty

will be

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because the sum of the given infinite series can be found using the derivative of a geometric series.

828 Easy

828 of 998

Application +4 | -1

If the ratio between the sums of

n

terms of two A.P.'s is

3 n+8: 7 n+15

, then the ratio between their 12th terms is

View answer and explanation

Correct option: B

7: 16

Option B (7:16) is correct because the ratio between their 12th terms is obtained by replacing n by (2n-1) in the given ratio, then substituting n=12.

829 Easy

829 of 998

Application +4 | -1

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

. are in A.P. such that

a_{4}-a_{7}+a_{10}=m

, then the sum of first 13 terms of this A.P., is :

View answer and explanation

Correct option: C

13 \mathrm{m}

Option C (13m) is correct because the sum of the first 13 terms of the A.P. can be expressed as 13 times the 7th term, which is given as m.

830 Easy

830 of 998

Application +4 | -1

If sum of the first 21 terms of the series

{\log _{{9^{1/2}}}}x + {\log _{{9^{1/3}}}}x + {\log _{{9^{1/4}}}}x + .....,

where

x > 0

is 504, then

x

is equal to

View answer and explanation

Correct option: D

Option D (81) is correct because after simplifying the logarithmic series and using the given sum, we find x = 81.

831 Easy

831 of 998

Analyzing +4 | -1

y=a^{\frac{1}{1-\log _a x}}

and

z=a^{\frac{1}{1-\log _a y}},

then

x

is equal to

View answer and explanation

Correct option: C

a^{\frac{1}{1-\log _a z}}

Option C (

a^{\frac{1}{1-\log _a z}}

) is correct because by substituting and simplifying the given logarithmic equations, we can express x in terms of a and z.

832 Easy

832 of 998

Application +4 | -1

The sum of 10 terms of the series

\sqrt{2}+\sqrt{6}+\sqrt{18}+\ldots

View answer and explanation

Correct option: A

121(\sqrt{6}+\sqrt{2})

Option A (

121(\sqrt{{6}}+\sqrt{{2}})

) is correct because the sum of the 10 terms of the given GP can be calculated using the formula for the sum of a GP, and simplifying the expression.

833 Easy

833 of 998

Application +4 | -1

If the 2 nd, 5th and 9th terms of a non-eustant AP are in GP, then the common ratio of this GP is

View answer and explanation

Correct option: D

\frac{4}{3}

Option D (

\frac{4}{3}

) is correct because the common ratio of the GP is derived by using the given condition that the 2nd, 5th, and 9th terms of the AP are in GP.

834 Easy

834 of 998

Application +4 | -1

p\mathrm{th},q\mathrm{th}

and

r\mathrm{th}

terms of a G.P. are

x,y,z

respectively, then

{x^{q - r}}{y^{r - p}}{z^{p - q}}

is equal to

View answer and explanation

Correct option: B

Option B (1) is correct because the expression simplifies to 1 using the properties of exponents and geometric progressions.

835 Easy

835 of 998

Application +4 | -1

The sum to infinity of the progression

9 - 3 + 1 - \frac{1}{3} + ...

View answer and explanation

Correct option: C

27 / 4

Option C (

27 / 4

) is correct because the sum to infinity of a GP is a/(1-r), where a=9 and r=-1/3.

836 Easy

836 of 998

Understanding +4 | -1

\underbrace{666 \ldots \ldots \ldots .6}_{\text {atimes }}

is equal to

View answer and explanation

Correct option: B

\frac{2}{3}\left(10^{\mathrm{n}}-1\right)

Option B (

\frac{2}{3}(10^{\mathrm{n}}-1)

) is correct because the repeating 6's can be expressed as a geometric series.

837 Easy

837 of 998

Understanding +4 | -1

Statement I: Three numbers

a, b, c

are in A.P., then

b

is called the arithmetic mean of

a

and

c

.

Statement II: Three numbers

a, b, c

are in A.P. iff

2 b=a+c

.

Choose the correct option.

View answer and explanation

Correct option: C

Both are true

Option C (Both are true) is correct because both statements accurately describe the relationship between arithmetic mean and arithmetic progression.

838 Easy

838 of 998

Application +4 | -1

Number of integeral solution satisfying

x+\sqrt{3-x} \geq \sqrt{3-x}+3

View answer and explanation

Correct option: A

Option A (1) is correct because the only integer solution that satisfies the given inequality and the domain restriction is x=3.

839 Easy

839 of 998

Application +4 | -1

The value of

x+y+z

is 15 if

a, x, y, z, b

are in A.P. while the value of

\frac{1}{x}+\frac{1}{y}+\frac{1}{z}

\frac{5}{3}

a, x, y, z, b

are in H.P. Then the value of a and

b

are

View answer and explanation

Correct option: B

1 and 9

Option B (1 and 9) is correct because it satisfies both the arithmetic and harmonic progression conditions given in the problem.

840 Easy

840 of 998

Application +4 | -1

5^2 \cdot 5^4 \cdot 5^6 \cdots 5^{2 n}=(0.04)^{-28}

, then

n

is equal to

View answer and explanation

Correct option: A

Option A (7) is correct because simplifying the equation and solving for n yields n = 7.

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