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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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991 Easy

991 of 998

Analyzing +4 | -1

For

0<0<\frac{\pi}{2}

, if

x = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}} \theta ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}} \phi ,z = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}} \theta {\sin ^{2n}}\phi ,

then

View answer and explanation

Correct option: B

x y=z x+z y-z

Option B (

x y=z x+z y-z

) is correct because it is derived from the relationships between x, y, and z, which are sums of infinite geometric series involving cosine and sine functions.

992 Easy

992 of 998

Application +4 | -1

Let

\alpha ,\beta ,\gamma \;\mathrm{and}\;\delta

are four positive real numbers such that their product is unity, then the least value of

\left( {1 + \mathrm{\alpha }} \right)\left( {1 + \mathrm{\beta }} \right)\left( {1 + \mathrm{\gamma }} \right)\left( {1 + \mathrm{\delta }} \right)

View answer and explanation

Correct option: B

Option B (16) is correct because, by applying the AM-GM inequality, the least value of the expression is found to be 16.

993 Easy

993 of 998

Application +4 | -1

The interior angles of a polygon are in AP. If the smallest angle is

120^{\circ}

and the common difference is

5^{\circ}

, then the number of sides is

View answer and explanation

Correct option: C

Option C (9) is correct because solving the equation for the sum of interior angles of a polygon in AP yields n=9 as the valid solution.

994 Easy

994 of 998

Application +4 | -1

The sum to infinity of the following series

\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\ldots \ldots

shall be

View answer and explanation

Correct option: B

Option B (1) is correct because the sum to infinity of the series can be expressed as a telescoping series, which converges to 1.

995 Easy

995 of 998

Application +4 | -1

Which of the following is harmonic mean of numbers

\frac{1}{3}

and

\frac{1}{7}

View answer and explanation

Correct option: B

\frac{1}{5}

Option B (1/5) is correct because the harmonic mean of 1/3 and 1/7 is calculated using the formula H = 2ab/(a+b).

996 Easy

996 of 998

Application +4 | -1

(1+3+5+\ldots+p)+(1+

3+5+\ldots+q)=(1+3+5+\ldots+r)

where each set of parentheses contains the sum of consecutive odd integers as shown, the smallest possible value of

p+q+r

, (where

p>6

) is

View answer and explanation

Correct option: B

Option B (21) is correct because the problem can be reduced to finding a Pythagorean triplet, and the smallest possible value of p+q+r is 21.

997 Easy

997 of 998

Application +4 | -1

S_n=n P+\frac{n(n-1)}{2} Q

, where

S_n

denotes the sum of the first

n

terms of an A.P., then the common difference is

View answer and explanation

Correct option: D

Q

Option D (

Q

) is correct because the common difference of the A.P. is derived by subtracting consecutive terms, which simplifies to

Q

998 Easy

998 of 998

Application +4 | -1

The sum of the series

1+2 x+3 x^2+4 x^3+\ldots

(where,

x

lies between 0 and 1 , i.e.

0 < x < 1

) is

View answer and explanation

Correct option: D

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

Option D,

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

, is the correct answer because it represents the sum of the given infinite series, derived by manipulating the series and using the formula for an infinite geometric series.

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