Let a < 1 , b < 1 , c < 1 and x = ∑ n = 0 ∞ a n , y = ∑ n = 0 ∞ b n , z = ∑ n = 0 ∞ c n . If 1 x , 1 y and 1 z are in Arithmetic Progression, then

a , b and c are in Arithmetic Progression.

a , 2 b and c are in Arithmetic Progression.

a , b and c satisfy 2 b 2 = a c .

a , b and c are in Geometric Progression.

MEDIUM

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The sum to infinity of the series $1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \frac{14}{3^{4}} + . \dots . .$ is

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Important Questions on Sequences and Series

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

If ${(20)}^{19} + 2 (21) {(20)}^{18} + 3 {(21)}^{2} {(20)}^{17} + . . . + 20 {(21)}^{19} = k {(20)}^{19}$ , then $k$ is equal to _____.

HARD

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

Let

S = 2 + \frac{6}{7} + \frac{12}{7^{2}} + \frac{20}{7^{3}} + \frac{30}{7^{4}} + \dots . .

then

4 S

is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

Let

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

be an A.P. If

\sum_{r = 1}^{\infty} \frac{a_{r}}{2^{r}} = 4

, then

4 a_{2}

is equal to ______.

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

For $k \in N$ , if the sum of the series $1 + \frac{4}{k} + \frac{8}{k^{2}} + \frac{13}{k^{3}} + \frac{19}{k^{4}} + . . . . . .$ is $10$ , then the value of $k$ is

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

The sum of the infinite series

1 + \frac{2}{3} + \frac{7}{3^{2}} + \frac{12}{3^{3}} + \frac{17}{3^{4}} + \frac{22}{3^{5}} + \dots \dots

is equal to:

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

Let

S = 109 + \frac{108}{5} + \frac{107}{5^{2}} + \dots . + \frac{2}{5^{107}} + \frac{1}{5^{108}}

. Then the value of

(16 S - (25)^{- 54})

is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

The sum

1 + 2 \cdot 3 + 3 \cdot 3^{2} + \dots \dots . . + 10 \cdot 3^{9}

is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

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>Arithmetico-geometric Progression

The sum of the series

\frac{13}{7^{2} . 2.1} + \frac{19}{7^{3} . 3.2} + \frac{25}{7^{4} . 4.3} + \dots . . . .

up to infinity is

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

If the value of ${(1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \dots . . upto \infty)}^{\log_{(0.25)} (\frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \dots . upto \infty)}$ is $l$ , then $l^{2}$ is equal to .

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

Given that

n

numbers of arithmetic means are inserted between two sets of numbers

a, 2 b

and

2 a, b

where

a, b \in R

. Suppose further that the

m^{th}

means between these sets of numbers are same, then the ratio

a : b

equals

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

The sum

\sum_{k = 1}^{20} k \frac{1}{2^{k}}

is equal to

HARD

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

e^{(\cos^{2} x + \cos^{4} x + \cos^{6} x + . . . . \infty) \log_{e} 2}

satisfies the equation

t^{2} - 9 t + 8 = 0

, then the value of

\frac{2 \sin x}{\sin x + \sqrt{3} \cos x}

, where

0 < x < \frac{π}{2}

, is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

Suppose

a_{1}, a_{2}, 2, a_{3}, a_{4}

be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is

2

and the sum of all

5

terms of the arithmetico-geometric progression is

\frac{49}{2}

, then

a_{4}

is equal to ______________

EASY

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \dots

\infty = \frac{π^{2}}{6}

then the value of

\frac{1}{1^{2}} + \frac{1}{3^{2}} + \frac{1}{5^{2}} + \dots

\infty

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

The sum of infinite number of terms of a Geometric Progression is

3

and the sum of the squares of its terms is

\frac{9}{2}

. The sum of the cubes of its terms is

EASY

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

Let

|a| < 1, |b| < 1, |c| < 1

and

x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n}

. If

\frac{1}{x}, \frac{1}{y} and \frac{1}{z}

are in Arithmetic Progression, then

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

S = \frac{7}{5} + \frac{9}{5^{2}} + \frac{13}{5^{3}} + \frac{19}{5^{4}} + \dots,

then

160 S

is equal to .

HARD

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

The sum of the series

\frac{1^{2}}{1 \cdot 2} + \frac{1^{2} + 2^{2}}{2 \cdot 3} + \frac{1^{2} + 2^{2} + 3^{2}}{3 \cdot 4} + \dots

upto

20

terms is

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetico-geometric Progression

{(10)}^{9} + 2 {(11)}^{1} {(10)}^{8} + 3 {(11)}^{2} {(10)}^{7} + ...... + 10 {(11)}^{9} = k {(10)}^{9},

then

k

is equal to :

The sum to infinity of the series 1+23+632+1033+1434+.….. is

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

The sum to infinity of the series $1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \frac{14}{3^{4}} + . \dots . .$ is