Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-1>Q 51

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If $p, q$ be the A.M., G.M., respectively between any two rational numbers $a$ and $b,$ then $p - q$ is equal to

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Important Questions on Sequences and Series

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-1>Q 52

The sum of the series

S = 1^{2} - 2^{2} + 3^{2} - 4^{2} + \dots - 2002^{2} + 2003^{2}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-1>Q 53

The sum

\sum_{r = 2}^{\infty} \frac{1}{r^{2} - 1}

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-1>Q 54

If

H_{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots \dots \dots . + \frac{1}{n},

then value of

1 + \frac{3}{2} + \frac{5}{3} + \dots \dots \dots + \frac{2 n - 1}{n}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-1>Q 55

Statement 1: The sum of the first 30 terms of the sequence $1, 2, 4, 7, 11, 16, 22, \dots \dots,$ is $4520$

Statement 2: If the successive differences of the terms of a sequence form an A.P., then general term of sequence is of the form $a n^{2} + b n + c$

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 2

Given the sequence of numbers

x_{1}, x_{2}, x_{3}, \dots \dots x_{2013}

which satisfies

\frac{x_{1}}{x_{1} + 1} = \frac{x_{2}}{x_{2} + 3} = \frac{x_{3}}{x_{3} + 5} = . . . . = \frac{x_{2013}}{x_{2013} + 4025}

, nature of sequence is

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 3

Suppose

a, b, c

are in A.P. and

a^{2}, b^{2}, c^{2}

are in G.P. if

a < b < c

and

a + b + c = \frac{3}{2},

then the value of

a

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 4

If

1, 2, 3 \dots

are first terms;

1, 3, 5 \dots .

are common differences and

S_{1}, S_{2}, S_{3} \dots

are sums of

n

terms of given

p

A.P.s, then

S_{1} + S_{2} + S_{3} + \dots + S_{p}

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 5

If the sum of

n

terms of a G.P. (with common ratio

r

) beginning with the

p^{th}

term is

k

times the sum of an equal number of terms of the same series beginning with the

q^{th}

term, then the value of

k

is