Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Archives>Q 15

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Let $A$ be the sum of the first $20$ terms and $B$ be the sum of the first $40$ terms of the series $1^{2} + 2 \cdot 2^{2} + 3^{2} + 2 \cdot 4^{2} + 5^{2} + 2 \cdot 6^{2} + \dots \dots$ If $B - 2 A = 100 λ,$ then $λ$ is equal to

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Important Questions on Progression and Series

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Archives>Q 16

Let

a_{1}, a_{2}, a_{3}, \dots, a_{49}

be in

A . P .

such that

\sum_{k = 0}^{12} a_{4 k + 1} = 416

and

a_{9} + a_{43} = 66

. If

a_{1}^{2} + a_{2}^{2} + \dots + a_{17}^{2} = 140 m,

then

m

is equal to

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Archives>Q 1

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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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Archives>Q 2

The value of

\sum_{k = 1}^{13} \frac{1}{\sin (\frac{π}{4} + \frac{(k - 1) π}{6}) \sin (\frac{π}{4} + \frac{k π}{6})}

is equal to

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Archives>Q 3

Let

b_{i} > 1

for

i = 1, 2, . . ., 101 .

Suppose

\log_{e} b_{1}, \log_{e} b_{2, . . . . . .,}

\log_{e} b_{101}

are in arithmetic progression

(A . P .)

with the common difference

\log_{e} 2

. Suppose

a_{1}, a_{2}, . . . ., a_{101}

are in

A . P .

such that

a_{1} = b_{1}

and

a_{51} = b_{51}

. If

t = b_{1} + b_{2} + . . + b_{51}

and

s = a_{1} + a_{2} + . . . + a_{51},

then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Archives>Q 1

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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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Archives>Q 1

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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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Archives>Q 2

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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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Archives>Q 3

Let

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

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

.