$\begin{array}{*{20}{c}} {{{\left( {666 \ldots .6} \right)}^2}}\\ {n - {\rm{digits}}} \end{array}\begin{array}{*{20}{c}} {\; + \;\left( {888 \ldots .8} \right)}\\ {n - {\rm{digits}}} \end{array}$ is equal to

$\frac{4}{9}\left( {{{10}^n} - 1} \right)$

$\frac{4}{9}{\left( {{{10}^n} - 1} \right)^2}$

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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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Showing 811-840 of 998 questions

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811 Easy

811 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.

812 Medium

812 of 998

Application +4 | -1

y = 1 + x + {x^2} + {x^3} + ...

then

x

is equal to

View answer and explanation

Correct option: A

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

Option A is correct because x can be expressed as (y-1)/y by recognizing the geometric series and solving for x.

813 Easy

813 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.

814 Medium

814 of 998

Application +4 | -1

Let

{S_k} = \frac{{1 + 2 + 3 + ... + k}}{k}.\mathrm{ }\,\,\mathrm{If}

S_1^2 + S_2^2 + ... + S_{10}^2 = \frac{5}{{12}}A,

then A is equal to

View answer and explanation

Correct option: D

303

Option C (

2\int_0^a {f\left( x \right)dx}

) is correct because using the properties of definite integrals and the given conditions, the integral simplifies to this expression.

815 Medium

815 of 998

Application +4 | -1

The sum of integers from 1 to 100 that are divisible by 2 or 5 is

View answer and explanation

Correct option: B

3050

Option B (3050) is correct because it's the sum of numbers divisible by 2 or 5, calculated using the principle of inclusion-exclusion.

816 Easy

816 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.

817 Medium

817 of 998

Application +4 | -1

Sum of

n

terms of the series

\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+\ldots

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it is the sum of the given arithmetic series.

818 Easy

818 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.

819 Medium

819 of 998

Analyzing +4 | -1

\begin{array}{cccccccccccccccccccc} {{{\left( {666 \ldots .6} \right)}^2}}\\ {n - \mathrm{digits}} \end{array}\begin{array}{cccccccccccccccccccc} {\; + \;\left( {888 \ldots .8} \right)}\\ {n - \mathrm{digits}} \end{array}

is equal to

View answer and explanation

Correct option: B

\frac{4}{9}\left( {{{10}^{2n}} - 1} \right)

Option B (

\frac{4}{9}\left( {{{10}^{2n}} - 1} \right)

) is correct because the expression simplifies to this result after converting the repeating digits to a geometric series and applying algebraic manipulations.

820 Easy

820 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.

821 Easy

821 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.

822 Medium

822 of 998

Understanding +4 | -1

The sum

1(1 !)+2(2 !)+3(3 !)+\ldots+n(n !)

equals

View answer and explanation

Correct option: C

(n+1) !-1

Option C (

(n+1) !-1

) is correct because by rewriting each term as ((k+1) - 1)k! and telescoping the sum, we arrive at (n+1)! - 1.

823 Easy

823 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.

824 Medium

824 of 998

Analyzing +4 | -1

If three real numbers

a,b,c

are in harmonic progression, then which of the following is true?

View answer and explanation

Correct option: B

\frac{1}{{bc}},\frac{1}{{ca}},\frac{1}{{ab}}

are in HP

Option C (

ab,bc,ca

are in HP) is incorrect. Option D (

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

are in HP) is incorrect. Option A (

\frac{1}{a},b,\frac{1}{c}

are in AP) is incorrect. Option B (

\frac{1}{{bc}},\frac{1}{{ca}},\frac{1}{{ab}}

are in HP) is correct because if

a, b, c

are in HP, then

bc, ca, ab

are in AP, and thus

\frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab}

are in HP.

825 Medium

825 of 998

Application +4 | -1

If the sides of a right angled triangle are in GP, then the cosine of the greater acute angle is

View answer and explanation

Correct option: A

\frac{2}{1+\sqrt{5}}

Option A (\frac{{2}}{{1+

\sqrt{{5}}

}}}) is correct because it is the cosine of the greater acute angle when the sides of a right-angled triangle are in GP.

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826 Medium

826 of 998

Application +4 | -1

If in a GP of 3 nterms,

S_1

denotes the sum of the first

n

terms,

S_2

denotes the sum of the second block of

n

terms and

\dot{S}_3

denotes the sum of the last

n

terms, then

S_1, S_2

and

S_3

are in

View answer and explanation

Correct option: B

Option B (GP) is correct because the sums of consecutive blocks of n terms in a GP also form a GP.

827 Easy

827 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.

828 Medium

828 of 998

Analyzing +4 | -1

a,\;b,\;c

are three distinct positive real numbers which are in HP, then

\frac{{3a + 2b}}{{2a - b}} + \frac{{3c + 2b}}{{2c - b}}

View answer and explanation

Correct option: A

Greater than or equal to 10

Option A (Greater than or equal to 10) is correct because the expression simplifies to 10 plus a positive term, making it greater than 10.

829 Medium

829 of 998

Application +4 | -1

0.272727 \ldots, x

and

0.727272 \ldots

are in HP, then

x

must be

View answer and explanation

Correct option: A

rational

Option A (rational) is correct because the Harmonic Mean of two rational numbers is always a rational number.

830 Easy

830 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.

831 Medium

831 of 998

Application +4 | -1

S

be the sum,

P

be the product and

R

be the sum of the reciprocals of

n

terms of a GP, then

{P^2}

is equal to

View answer and explanation

Correct option: A

{\left( {\frac{S}{R}} \right)^n}

Option A (

{( \frac{S}{R} )}^n

) is correct because it is derived by expressing S, P, and R in terms of the first term and common ratio of the G.P. and then verifying the relationship.

832 Easy

832 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.

833 Medium

833 of 998

Application +4 | -1

The sum of

n

terms of the series

\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\ldots

View answer and explanation

Correct option: B

\frac{-1+\sqrt{2 n+1}}{2}

Option B (

\frac{-1+\sqrt{2 n+1}}{2}

) is correct because it is the sum of the given series after rationalizing and simplifying.

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 Medium

835 of 998

Application +4 | -1

If the sum of

n

terms of an AP is

3 n^2+5 n

, then which of its terms is 164 ?

View answer and explanation

Correct option: B

27 th

Option B (27 th) is correct because it is the term number that has a value of 164 in the given arithmetic progression.

836 Easy

836 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.

837 Easy

837 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.

838 Easy

838 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.

839 Easy

839 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.

840 Easy

840 of 998

Analyzing +4 | -1

If both the regression coefficients

b_{y x}

and

b_{x y}

are positive, then

View answer and explanation

Correct option: A

\frac{1}{b_{y x}}+\frac{1}{b_{x y}} > \frac{2}{r}

Option A (

\frac{1}{b_{y x}}+\frac{1}{b_{x y}} > \frac{2}{r}

) is correct because the geometric mean of regression coefficients is greater than or equal to the harmonic mean.

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