EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Main - 28 June 2022 Shift 2>Q 142

MEDIUM

JEE Main

IMPORTANT

Earn 100

If $n$ arithmetic means are inserted between a and $100$ such that the ratio of the first mean to the last mean is $1 : 7$ and $a + n = 33$ , then the value of $n$ is

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Important Questions on Sequences and Series

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Main - 28 June 2022 Shift 2>Q 143

Let for

n = 1, 2, \dots \dots, 50, S_{n}

be the sum of the infinite geometric progression whose first term is

n^{2}

and whose common ratio is

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

. Then the value of

\frac{1}{26} + \sum_{n = 1}^{50} (S_{n} + \frac{2}{n + 1} - n - 1)

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Main - 29 June 2022 Shift 1>Q 144

Let

{\{a_{n}\}}_{n = 0}^{\infty}

be a sequence such that

a_{0} = a_{1} = 0

and

a_{n + 2} = 2 a_{n + 1} - a_{n} + 1

for all

n \geq 0

. Then,

\sum_{n = 2}^{\infty} \frac{a_{n}}{7^{n}}

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Main - 29 June 2022 Shift 2>Q 145

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}} + \dots

is equal to:

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Main - 29 June 2022 Shift 2>Q 146

Let

3, 6, 9, 12, \dots

upto

78

terms and

5, 9, 13, 17, \dots

upto

59

terms be two series. Then, the sum of the terms common to both the series is equal to ______.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Main - 25 July 2022 Shift 1>Q 147

Let

a, b

be two non-zero real numbers. If

p

and

r

are the roots of the equation

x^{2} - 8 a x + 2 a = 0

and

q

and

s

are the roots of the equation

x^{2} + 12 b x + 6 b = 0

, such that

\frac{1}{p}, \frac{1}{q}, \frac{1}{r}, \frac{1}{s}

are in A.P., then

a^{- 1} - b^{- 1}

is equal to _____ .

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Main - 25 July 2022 Shift 1>Q 148

Let

a_{1} = b_{1} = 1, a_{n} = a_{n - 1} + 2

and

b_{n} = a_{n} + b_{n - 1}

for every natural number

n \geq 2

. Then

\sum_{n = 1}^{15} a_{n} \cdot b_{n}

is equal to _____ .

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Main - 25 July 2022 Shift 2>Q 149

The sum

\sum_{n = 1}^{21} \frac{3}{(4 n - 1) (4 n + 3)}

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Main - 26 July 2022 Shift 1>Q 150

Consider two G.Ps.

2, 2^{2}, 2^{3}, \dots

and

4, 4^{2}, 4^{3}, \dots

of

60

and

n

terms respectively. If the geometric mean of all the

60 + n

terms is

{(2)}^{\frac{225}{8}}

, then

\sum_{k = 1}^{n} k (n - k)

is equal to: