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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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871 Easy

871 of 998

Application +4 | -1

Suppose

m

arithmetic means are inserted between 1 and 31. If the ratio of the second mean to the

m

th mean is

1: 4

, then

m

is equal to

View answer and explanation

Correct option: B

Option B (9) is correct because by using the properties of arithmetic means and the given ratio, we can solve for the number of means inserted.

872 Easy

872 of 998

Analyzing +4 | -1

For what value of

b,

will the roots of the equation

\cos x = b, - 1 \le b \le 1

when arranged in ascending order of their magnitudes, form an A.P.?

View answer and explanation

Correct option: A

- 1

Option A (

-1

) is correct because when

\cos x = -1

, the solutions for x form an arithmetic progression.

873 Medium

873 of 998

Understanding +4 | -1

If the sum of the first

n

terms of a sequence is of the form

{A} n^2+{B} n

where

{A}, {B}

are constants independent of

n

, then the sequence is

View answer and explanation

Correct option: A

an A.P.

Option D (6) is correct because the number of common roots is equal to the greatest common divisor (GCD) of 12 and 18, which is 6.

874 Easy

874 of 998

Application +4 | -1

{a^x} = b,\;{b^y} = c,\;{c^z} = a,

, then value of

xyz

View answer and explanation

Correct option: B

Option B (1) is correct because by substituting and simplifying the given equations, we find that

xyz = 1

875 Medium

875 of 998

Application +4 | -1

a, b

and

c

are in HP, then

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

and

\frac{c}{a+b}

are in

View answer and explanation

Correct option: C

Option C (HP) is correct because if a, b, c are in HP, then their reciprocals are in AP, and manipulating the AP sequence leads to the conclusion that the given terms are also in HP.

876 Easy

876 of 998

Application +4 | -1

In an arithmetic progression, the 24th term is 100. Then, the sum of the first 47 terms of the arithmetic progression is

View answer and explanation

Correct option: D

4700

Option D (4700) is correct because the sum of the first 47 terms is calculated using the given 24th term and the arithmetic progression formula.

877 Easy

877 of 998

Application +4 | -1

The sum of the series

S=\sum\limits_{{r=1}}^n \log \left(\frac{a^{r+1}}{b^{r-1}}\right)

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it represents the simplified sum of the given logarithmic series.

878 Easy

878 of 998

Application +4 | -1

If the sum of first

n

terms of an AP is

c{n^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){c^2}}}{3}

Option C is correct because it represents the sum of the squares of the terms of the AP, derived from the given sum of the first n terms.

879 Easy

879 of 998

Application +4 | -1

The sum of the series

6 + 66 + 666 + ...

upto

n

term is

View answer and explanation

Correct option: B

\frac{{2({{10}^{n + 1}} - 9n - 10)}}{{27}}

Option B is correct because the sum of the series is derived by manipulating the series and using the formula for the sum of a geometric progression.

880 Medium

880 of 998

Application +4 | -1

2^{10}+2^9 \cdot 3^1+2^8 \cdot 3^2+\ldots+2.3^9+3^{10}

=S-2^{11}

, then

S

is equal to

View answer and explanation

Correct option: A

3^{11}

Option A (

3^{11}

) is correct because it is the result of summing the given series and adding

2^{11}

to it.

881 Medium

881 of 998

Application +4 | -1

If sum of all terms of an infinite GP is

\frac{1}{5}

times the sum of odd terms. Then, common ratio is

View answer and explanation

Correct option: C

-\frac{4}{5}

Option C (

-\frac{{4}}{{5}}

) is correct because by setting up the equation relating the sum of the infinite GP to the sum of its odd terms, we can solve for the common ratio, which is -4/5.

882 Easy

882 of 998

Application +4 | -1

7^{\text {th }}

term of an A.P. 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 the first 13 terms can be directly calculated using the 7th term, which is the middle term.

883 Easy

883 of 998

Application +4 | -1

(0.16)^{\log _{2.5}\left[(1 / 3)+\left(1 / 3^2\right)+\cdots+\infty\right]}=

View answer and explanation

Correct option: D

Option D (4) is correct because after simplifying the geometric series and using logarithm properties, the expression evaluates to 4.

884 Easy

884 of 998

Remembering +4 | -1

Consider the following statements.

I. The

n^{\text {th}}

term of a G.P. with first term '

a

' and common ratio '

r

' is given by

a_n=a \cdot r^{n-1}

.

II. Geometric mean of

{a}

and

{b}

is given by

({ab})^{1 / 3}

Choose the correct option.

View answer and explanation

Correct option: A

Only I is true

Option A (Only I is true) is correct because the formula for the nth term of a G.P. is correctly stated, while the geometric mean formula is incorrect.

885 Medium

885 of 998

Application +4 | -1

In the sequence

1,2,2,4,4,4

4,8,8,8,8,8,8, \ldots

, where

n

consecutive terms have the value

n

, then 1025 th term is

View answer and explanation

Correct option: B

2^{10}

Option B (

2^{10}

) is correct because the 1025th term falls within the block of

2^{10}

consecutive terms having the value

2^{10}

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886 Easy

886 of 998

Application +4 | -1

a+c, a+b, b+c

are in G.P. and

a, c, b

are in H.P. where

a, b, c>0

, then the value of

\frac{a+b}{c}

is:

View answer and explanation

Correct option: B

Option B (2) is correct because by using the properties of Geometric Progression (GP) and Harmonic Progression (HP) and solving the resulting equations, we find that

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

887 Easy

887 of 998

Application +4 | -1

Let

x=\sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \ldots \text { upto } \infty}}}}

and

y=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\ldots \text { upto } \infty}}}}

, then

x-y

equals

View answer and explanation

Correct option: A

Option A (0) is correct because both x and y simplify to 2, making their difference zero.

888 Medium

888 of 998

Application +4 | -1

Let

x, y, z \in R^{+}

such that

x+y+z=27

. If maximum value of

x^2 y^3 z^4

\lambda \cdot 6^{10}

, then find the value of

\lambda

View answer and explanation

Correct option: A

Option A (9) is correct because the maximum value of the expression is found using AM-GM inequality, resulting in

\lambda = 9

889 Easy

889 of 998

Application +4 | -1

a_1, a_2, \ldots, a_{10}

are in G.P. such that

\frac{a_1}{a_5}=25

, then

\frac{a_{10}}{a_8}=

View answer and explanation

Correct option: A

Option A (25) is correct because in a G.P., the ratio of terms with a constant difference in their indices is constant.

890 Easy

890 of 998

Application +4 | -1

The minimum value of the quantity

\frac{\left(a^2+3 a+1\right)\left(b^2+3 b+1\right)\left(c^2+3 c+1\right)}{a b c}

, where

a, b, c \in R^{+}

, is

View answer and explanation

Correct option: B

125

Option B (125) is correct because by applying AM >= GM inequality to each factor, we find the minimum value of the expression.

891 Easy

891 of 998

Application +4 | -1

If sum of an infinite geometric series is

\frac{4}{3}

and its 1st term is

\frac{3}{4}

, then its common ration is

View answer and explanation

Correct option: A

\frac{7}{{16}}

Option A (7/16) is correct because it is the common ratio calculated using the formula for the sum of an infinite geometric series.

892 Medium

892 of 998

Analyzing +4 | -1

{5^{3{x^2}{{\log }_{10}}2}} = {2^{\left( {x + \frac{1}{2}} \right){{\log }_{10}}25}},

then

x

equals to

View answer and explanation

Correct option: A

1,\; - \frac{1}{3}

Option A (

1, -\frac{1}{3}

) is correct because these are the solutions obtained by equating the exponents after taking logarithms and solving the resulting quadratic equation.

893 Medium

893 of 998

Application +4 | -1

If 6 th term of a GP is 32 and its 8 th term is 128 , then the value of the common ratio is

View answer and explanation

Correct option: B

Option B (2) is correct because the common ratio can be found by dividing the 8th term by the 6th term and taking the square root.

894 Easy

894 of 998

Application +4 | -1

a_1, a_2, a_3, . . . , a_n, . . .

are in A.P. such that

a_4 - a_7 + a_{10} = m,

then sum of the first 13 terms of the A.P. is

View answer and explanation

Correct option: C

13 \,m

Option C (

13m

) is correct because the given condition simplifies to relate to the 7th term, and the sum of the first 13 terms is 13 times the 7th term.

895 Easy

895 of 998

Application +4 | -1

a, b, c

are in A.P. then the points

(a, x),(b, y)

and

(c, z)

are collinear if

View answer and explanation

Correct option: D

x, y, z

are in A.P.

Option D (

x, y, z

are in A.P.) is correct because the collinearity condition implies a constant slope, leading to

x, y, z

being in arithmetic progression.

896 Medium

896 of 998

Application +4 | -1

The sum of all odd numbers of four digits which are divisible by 9 , is

View answer and explanation

Correct option: A

2754000

Option A (2754000) is correct because it is the sum of the arithmetic progression of four-digit odd numbers divisible by 9.

897 Medium

897 of 998

Application +4 | -1

\lim\limits_{x\, \to \,2} \left( {\sum\limits_{n\, = \,1}^9 {\frac{x}{{n\left( {n + 1} \right){x^2} + 2\left( {2n + 1} \right)x + 4}}} } \right)

is equal to

View answer and explanation

Correct option: A

\frac{9}{{44}}

Option A (

\frac{9}{{44}}

) is correct because it is the value of the limit after simplifying the summation using partial fractions.

898 Medium

898 of 998

Analyzing +4 | -1

The sum of the series

\frac{1}{{2!}} + \frac{1}{{4!}} + \frac{1}{{6!}} + \ldots

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the series can be expressed in terms of e and simplified to this expression.

899 Easy

899 of 998

Application +4 | -1

If in a geometric progression

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

and

S_n=189

then the value of

n

is:

View answer and explanation

Correct option: B

Option B (6) is correct because the number of terms is found by using the formulas for the nth term and the sum of a geometric progression.

900 Easy

900 of 998

Application +4 | -1

{\log _{10}}343 = 2.5353,

then the least positive integer n such that

{7^n} > {10^5},

View answer and explanation

Correct option: B

Option B (6) is correct because it is the smallest integer value of n that satisfies the inequality

{7^n} > {10^5}

based on the given logarithm value.

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