Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Exercises>Q 34

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If $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in H.P. Then, which is of the following is/are possible?

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Important Questions on Progression and Series

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Exercises>Q 35

The corresponding first and the

{(2 n - 1)}^{th}

terms of an A.P., a G.P. and an H.P. are equal. If their

n^{th}

terms are

a, b

and

c

respectively, then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Exercises>Q 36

Let

E = \frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + . . . .,

Then,

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

Let

a, b, c, d

be four distinct real numbers in

A . P .

Then the smallest positive value of

k

satisfying

2 (a - b) + k {(b - c)}^{2} + {(c - a)}^{3} = 2 (a - d) + {(b - d)}^{2} + {(c - d)}^{3} i s

HARD

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

Let the fourth term of an arithmetic progression be

6

and

m^{th}

term be

18 .

If

A . P .

has integral terms only then the numbers of such

A . P .

s

is

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

The

5^{th}

and

8^{th}

terms of a geometric sequence of real numbers are

7!

and

8!

, respectively. If sum of first

n

terms of the

G . P .

is

2205,

then

n

equals

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Exercises>Q 4

Let

a_{1}, a_{2}, a_{3}, . . . ., a_{101}

are in

G . P .

with

a_{101} = 25 and \sum_{i = 1}^{201} a_{i} = 625 .

Then the value of

\sum_{i = 1}^{201} \frac{1}{a_{i}}

equals

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

For

a, b > 0,

let

5 a - b, 2 a + b, a + 2 b

be in

A . P .

and

{(b + 1)}^{2}, a b + 1, {(a - 1)}^{2}

are in

G . P .

, then the value of

(a^{- 1} + b^{- 1})

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 5 - Progression and Series>Exercises>Q 6

Let

a + a r_{1} + a {r_{1}}^{2} + . . . . + \infty and a + a r_{2} + a {r_{2}}^{2} + . . . . + \infty

be two infinite series of positive numbers with the same first term. The sum of the first series is

r_{1}

and the sum of the second series is

r_{2} .

Then the value of

(r_{1} + r_{2})

is