HARD

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The sum of the first $10$ terms of an A.P. is $155$ and the sum of the first two terms of a G.P. is $9$ . Determine the progressions if the first term of the A.P. is equal tot eh common ratio of the G.P. and the first term of the G.P. is equal to the common difference of the A.P.

Important Questions on Sequences and Series

MEDIUM

Mathematics>Algebra>Sequences and Series>Geometric Progressions

Let

a, b

and

c

be the

2 x^{2} - 29 x + 50 = 0

and

2 x^{2} - 4 x - 25 x + 50 = 0

terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then

\frac{a}{c}

is equal to:

MEDIUM

Mathematics>Algebra>Sequences and Series>Geometric Progressions

Let

C_{0}

be a circle of radius

1

. For

n \geq 1,

let

C_{n}

be a circle whose area equals the area of a square inscribed in

C_{n - 1}

. Then

(\sum_{u = 0}^{\infty} area (C_{i}))

equals,

MEDIUM

Mathematics>Algebra>Sequences and Series>Geometric Progressions

If three distinct numbers

a, b, c

are in G.P. and the equations

a x^{2} + 2 b x + c = 0

and

d x^{2} + 2 e x + f = 0

have a common root, then which one of the following statements is correct?

EASY

Mathematics>Algebra>Sequences and Series>Geometric Progressions

Let

a_{1}, a_{2}, {\dots . . a}_{10}

be a G.P. If

\frac{a_{3}}{a_{1}} = 25,

then

\frac{a_{9}}{a_{5}}

equals:

HARD

Mathematics>Algebra>Sequences and Series>Geometric Progressions

If $m$ is the $A . M .$ of two distinct real numbers $I$ and $n$ $(I, n > 1)$ and $G_{1}, G_{2} and G_{3}$ are three geometric means between $I and n$ , then $G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}$ equals

MEDIUM

Mathematics>Algebra>Sequences and Series>Geometric Progressions

α, β

and

γ

are three consecutive terms of a non-constant G.P. Such that the equations

α x^{2} + 2 β x + γ = 0

and

x^{2} + x - 1 = 0

have a common root, then

α (β + γ)

is equal to:

MEDIUM

Mathematics>Algebra>Sequences and Series>Geometric Progressions

If the

2^{n d}, 5^{t h}

and

9^{t h}

terms of a non-constant arithmetic progression are in geometric progression, then the common ratio of this geometric progression is

EASY

Mathematics>Algebra>Sequences and Series>Geometric Progressions

The product

2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \cdot . . . .

\infty

is equal to:

MEDIUM

Mathematics>Algebra>Sequences and Series>Geometric Progressions

The sum of the

3^{r d}

and the

4^{t h}

terms of a

G . P .

60

and the product of its first three terms is

1000.

If the first term of this

G . P .

is positive, then its

7^{t h}

term is:

MEDIUM

Mathematics>Algebra>Sequences and Series>Geometric Progressions

The quotient when

1 + x^{2} + x^{4} + . . . + x^{34}

is divided by

1 + x + x^{2} + . . . + x^{17}

EASY

Mathematics>Algebra>Sequences and Series>Geometric Progressions

The product of three consecutive terms of a

G . P .

512

. If

4

is added to each of the first and the second of these terms, the three terms now form an

A . P .

, then the sum of the original three terms of the given

G . P .

is :

HARD

Mathematics>Algebra>Sequences and Series>Geometric Progressions

The number of real solutions of the equation

\sin^{- 1} (\sum_{i = 1}^{\infty} x^{i + 1} - x \sum_{i = 1}^{\infty} {(\frac{x}{2})}^{i}) = \frac{π}{2} - \cos^{- 1} (\sum_{i = 1}^{\infty} {(- \frac{x}{2})}^{i} - \sum_{i = 1}^{\infty} {(- x)}^{i})

lying in the interval

(- \frac{1}{2}, \frac{1}{2})

is____.

(Here, the inverse trigonometric functions

\sin^{- 1} x & \cos^{- 1} x

assume values in

[- \frac{π}{2}, \frac{π}{2}] & [0, π]

respectively.)

EASY

Mathematics>Algebra>Sequences and Series>Geometric Progressions

If the sum of an infinite GP be

9

and sum of first two terms be

5

then their common ratio is …..

MEDIUM

Mathematics>Algebra>Sequences and Series>Geometric Progressions

Let

a, b

and

c

be in

G . P .

with common ratio

r,

where

a \neq 0

and

0 < r \leq \frac{1}{2} .

3 a, 7 b

and

15 c

are the first three terms of an

A . P .,

then the

4^{t h}

term of this

A . P .

is :

MEDIUM

Mathematics>Algebra>Sequences and Series>Geometric Progressions

The greatest positive integer

k,

for which

49^{k} + 1

is a factor of the sum

49^{125} + 49^{124} + \dots + 49^{2} + 49 + 1,

HARD

Mathematics>Algebra>Sequences and Series>Geometric Progressions

Let

a_{n}

be the

n^{t h}

term of a G.P. of positive terms. If

\sum_{n = 1}^{100} a_{2 n + 1} = 200

and

\sum_{n = 1}^{100} a_{2 n} = 100,

then

\sum_{n = 1}^{200} a_{n}

is equal to:

MEDIUM

Mathematics>Algebra>Sequences and Series>Geometric Progressions

x = \sum_{n = 0}^{\infty} {(- 1)}^{n} \tan^{2 n} θ

and

y = \sum_{n = 0}^{\infty} \cos^{2 n} θ,

for

0 < θ < \frac{π}{4},

then:

HARD

Mathematics>Algebra>Sequences and Series>Geometric Progressions

Let $S_{1}$ be the sum of areas of the squares whose sides are parallel to coordinate axes. Let $S_{2}$ be the sum of areas of the slanted squares as shown in the figure. Then $S_{1} / S_{2}$ is

Question Image

MEDIUM

Mathematics>Algebra>Sequences and Series>Geometric Progressions

The sum of first

20

terms of the sequence

0.7, 0.77, 0.777, . . . . . .,

is :

HARD

Mathematics>Algebra>Sequences and Series>Geometric Progressions

Let

b_{i} > 1

for

i = 1, 2, \dots ., 101 .

Suppose

\log_{e} b_{1}, \log_{e} b_{2}, \dots . ., \log_{e} b_{101}

are in Arithmetic Progression (A.P.) with the common difference

\log_{e} 2 .

Suppose

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

are in A.P. such that

a_{1} = b_{1}

and

a_{51} = b_{51} .

t = b_{1} + b_{2} + \dots + b_{51}

and

s = a_{1} + a_{2} + \dots + a_{51}

, then

The sum of the first 10 terms of an A.P. is 155 and the sum of the first two terms of a G.P. is 9. Determine the progressions if the first term of the A.P. is equal tot eh common ratio of the G.P. and the first term of the G.P. is equal to the common difference of the A.P.

Important Questions on Sequences and Series

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The sum of the first $10$ terms of an A.P. is $155$ and the sum of the first two terms of a G.P. is $9$ . Determine the progressions if the first term of the A.P. is equal tot eh common ratio of the G.P. and the first term of the G.P. is equal to the common difference of the A.P.