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

HARD

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$\frac{1}{y - x} + \frac{1}{y - z} = \frac{1}{x} + \frac{1}{z}$ , then $x, y, z$ are in ______.

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Important Questions on Progression and Series

MEDIUM

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

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

MEDIUM

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

a, b, c

are in arithmetic progression, then

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

will be in

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

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

are in

AP

, then

a, b, c

are in

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

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>Arithmetic>Progression and Series>Harmonic Progression

\cos (x - y), \cos x

and

\cos (x + y)

are in

HP

, then

\cos x \sec (y / 2)

is equal to

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

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

MEDIUM

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

x, y, z

are in

HP

, then

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

is equal to

MEDIUM

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

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

, where

x, y, z

are unequal positive numbers and

a, b, c

are in

GP

, then

HARD

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

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

MEDIUM

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

x

is the harmonic mean between

y

and

z

, then which one of the following is correct?

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

H

is the harmonic mean between

P

and

Q

then the value of

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

is :

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

Common difference of an A.P. is

5

and

9^{th}

term is

43

. Find the sum of

11^{th}

and

15^{th}

term of corresponding H.P.

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

If the ratio of harmonic mean of two numbers to their geometric mean is

12 : 13

, find the ratio of the numbers.

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

If the first two terms of a H.P. are

\frac{2}{5}

and

\frac{12}{13}

, respectively, which of the following terms is the largest term?

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

In and H.P, If

3^{r d}

term is

\frac{1}{3}

and

6^{t h}

term is

\frac{1}{5}

, Find

T_{4}

MEDIUM

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

H

is the harmonic mean of

P

and

Q

, then the value of

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

is_____.

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

Ratio of the

4^{th}

and

17^{th}

term of a H.P. is

1 : \frac{1}{4}

and common difference for corresponding A.P. is

3 .

Find the ratio of the

1^{st}

and

9^{th}

terms of H.P.

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

Determine the

5^{t h}

term and the

49^{t h^{}}

term of the harmonic progression

6, 4, 3 . . . . .

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

Determine the

5^{t h}

and

7^{t h}

term of the harmonic profression

\frac{1}{3}, \frac{1}{6}, \frac{1}{9} . . . . .

EASY

Mathematics>Arithmetic>Progression and Series>Harmonic Progression

If the arithmetic and the harmonic mean of two numbers are $4.5$ and $4$ , respectively, then one of the numbers is equal to:

1y-x+1y-z=1x+1z, then x, y, z are in ______.

Important Questions on Progression and Series

Learn and Prepare for any exam you want !

$\frac{1}{y - x} + \frac{1}{y - z} = \frac{1}{x} + \frac{1}{z}$ , then $x, y, z$ are in ______.