If x = ∑ n = 0 ∞ a n , y = ∑ n = 0 ∞ b n , z = ∑ n = 0 ∞ c n , where a , b , c are in A.P. and a < 1 , b < 1 , c < 1 , a b c ≠ 0 , then

1 x + 1 y + 1 z = 1 - a + b + c

EASY

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If a, b, c are in H.P., then the straight line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point and that point is

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Important Questions on Sequences and Series

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

x

is the harmonic mean between

y

and

z

, then which one of the following is correct?

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

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

, where

a, b, c

are in A.P. and

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

a b c \neq 0

, then

EASY

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

For two observations, the sum is S and product is P. What is the harmonic mean of these two observations?

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

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

EASY

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

a > 1, b > 1

and

c > 1

are in geometric progression, then

\frac{1}{1 + \log_{e} a}, \frac{1}{1 + \log_{e} b}, \frac{1}{1 + \log_{e} c}

will be in

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

Let

f : R \to R

be such that for all

x \in R (2^{1 + x} + 2^{1 - x}), f (x)

and

(3^{x} + 3^{- x})

are in A.P., then the minimum value of

f (x)

EASY

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

H

is the harmonic mean between

P

and

Q

then the value of

\frac{H}{P} + \frac{H}{Q}

is :

EASY

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

Let

x

and

y

be two positive real numbers such that

x + y = 1

. Then the minimum value of

\frac{1}{x} + \frac{1}{y}

is-

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

a, b, c

are in arithmetic progression, then

\frac{a}{b c}, \frac{1}{c}, \frac{2}{b}

will be in

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

x, y, z

are in

HP

, then

\log (x + z) + \log (x - 2 y + z)

is equal to

EASY

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

x_{1}, x_{2}, \dots . ., x_{n}

and

\frac{1}{h_{1}}, \frac{1}{h_{2}}, \dots . ., \frac{1}{h_{n}}

are two A.P.s such that

x_{3} = h_{2} = 8 & x_{8} = h_{7} = 20

, then

x_{5} \cdot h_{10}

is equal to

HARD

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

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

are in

HP

and

f (k) = \sum_{r = 1}^{n} a_{r} - a_{k}

, then

\frac{a_{1}}{f (1)}, \frac{a_{2}}{f (2)}, \frac{a_{3}}{f (3)}, \dots, \frac{a_{n}}{f (n)}

are in

EASY

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

\frac{(b + c - a)}{a}, \frac{(c + a - b)}{b}, \frac{(a + b - c)}{c}

are in

AP

, then

a, b, c

are in

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

a^{x} = b^{y} = c^{z}

, where

x, y, z

are unequal positive numbers and

a, b, c

are in

GP

, then

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

\cos (x - y), \cos x

and

\cos (x + y)

are in

HP

, then

\cos x \sec (y / 2)

is equal to

HARD

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

The number of solutions of the equation $\sin (e^{x}) = 5^{x} + 5^{- x}$ is –

HARD

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

ln (a + c), ln (c - a), ln (a - 2 b + c)

are in A.P., then

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

H_{1}, H_{2} a r e 2 H . M .^{'} s

between

a, b t h e n \frac{H_{1} + H_{2}}{H_{1} \cdot H_{2}} =

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

a, b, c

are in

H . P ., b, c, d

are in

G . P .

and

c, d, e

are in

A . P .

then value of

e

is -

HARD

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

The p^th term T_p of HP is q(p + q) and q^th term T_q is p (p + q) when p > 1, q > 1 (p and q are integers), then

If a, b, c are in H.P., then the straight line xa+yb+1c=0 always passes through a fixed point and that point is

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

If a, b, c are in H.P., then the straight line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point and that point is