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

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Suppose four distinct positive numbers $a_{1}, a_{2}, a_{3}, a_{4}$ are in G.P. Let $b_{1} = a_{1}, b_{2} = b_{1} + a_{2}, b_{3} = b_{2} + a_{3}$ and $b_{4} = b_{3} + a_{4}$ .

STATEMENT - $1$ : The numbers $b_{1}, b_{2}, b_{3}, b_{4}$ are neither in A.P. nor in G.P.

STATEMENT - $2$ : The numbers $b_{1}, b_{2}, b_{3}, b_{4}$ are in H.P.

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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-3>Q 6

If the sum of first

n

terms of an A.P. is

c n^{2}

, then the sum of squares of these

n

terms is

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 7

Let

S_{k}, k = 1, 2 . . . ., 100,

denote the sum of the infinite geometric series whose first term is

\frac{k - 1}{k!}

and the common ratio is

\frac{1}{k},

then the value of

\frac{100^{2}}{100!} + \sum_{k = 1}^{100} |(k^{2} - 3 k + 1) S_{k}|

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 8

Let

a_{1}, a_{2}, a_{3}, . . . ., a_{11}

be real numbers satisfying

a_{1} = 15, 27 - 2 a_{2} > 0

and

a_{k} = 2 a_{k - 1} - a_{k - 2}

for

k = 3, 4, . . ., 11

. If

\frac{a_{1}^{2} + a_{2}^{2} + . . . . + a_{11}^{2}}{11} = 90,

then the value of

\frac{a_{1} + a_{2} + . . . + a_{11}}{11}

is equal to

.

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 9

Let

a_{1}, a_{2}, a_{3}, \dots, a_{100}

be an arithmetic progression with

a_{1} = 3

and

S_{p} = \sum_{i = 1}^{p} a_{i}, 1 \leq p \leq 100

for any integer

n

with

1 \leq n \leq 20,

let

m = 5 n .

If

\frac{S_{m}}{S_{n}}

does not depend on

n,

then

a_{2}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 10

The minimum value of the sum of real numbers

a^{- 5}, a^{- 4}, 3 a^{- 3}, 1, a^{8}

and

a^{10}

, where

a > 0

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 11

Let

a_{1}, a_{2}, a_{3}, \dots .

be in harmonic progression with

a_{1} = 5

and

a_{20} = 25

. The least positive integer

n

for which

a_{n} < 0

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 12

Let

S_{n} = \sum_{k = 1}^{4 n} (- 1)^{\frac{k (k + 1)}{2}} k^{2} .

Then

S_{n}

can take value

(s)

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 13

A pack contains

n

cards numbered from

1

to

n .

Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is

1224 .

If the smaller of the numbers on the removed cards is

k,

then

k - 20 =

.