Mathematics>Algebra>Sequence and Series>Harmonic Progression

MEDIUM

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If $H$ is the harmonic mean of numbers $1, 2, 2^{2}, 2^{3}, \dots, 2^{n - 1}$ , then what is $n / H$ equal to?

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Important Questions on Sequence and Series

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

If

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

, where

x, y, z

are unequal positive numbers and

a, b, c

are in

GP

, then

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

If

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>Algebra>Sequence and Series>Harmonic Progression

If

\cos (x - y), \cos x

and

\cos (x + y)

are in

HP

, then

\cos x \sec (y / 2)

is equal to

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

If

a, b, c

are in

H . P .

and if

(\frac{a + b}{2 a - b}) + (\frac{c + b}{2 c - b}) > \sqrt{λ \sqrt{λ \sqrt{λ . .. \infty}}}

, then the value of

λ

must be

EASY

Mathematics>Algebra>Sequence and Series>Harmonic Progression

If the

(n + 1) th

term of a harmonic progression is twice the

(3 n + 1) th

term, find the ratio of the first term to the

(n + 1) th

term.

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

x + y + z = 15

, when

a, x, y, z, b

are in A.P. and

\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{5}{3}

when

a, x, y, z, b

are in H.P., then the quadratic equation whose roots are

\frac{1}{a}

and

\frac{1}{b}

is

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

If

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

are in A.P., then

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

If

x_{1}, x_{2}, x_{3}, . . . . . . . ., x_{2008}

are in

H . P .

and

Σ_{i = 1}^{2007} x_{i} x_{i + 1} = λ x_{1} x_{2008}

, then

λ

must be equal to:

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

If harmonic mean of

\frac{1}{2}, \frac{1}{2^{2}}, \frac{1}{2^{3}}, \dots \frac{1}{2^{10}}

is

\frac{λ}{2^{10} - 1}

, then

λ =

EASY

Mathematics>Algebra>Sequence and Series>Harmonic Progression

If

\frac{1}{2}, \frac{1}{4}

and

\frac{1}{6}

are three continuous terms of a series. Find the nature of the series.

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

If the

(m + 1)^{t h}, (n + 1)^{t h} & (r + 1)^{t h}

terms of an AP are in GP

& m, n, r

are in H P, then the ratio of the common difference to the first term of the AP is -

EASY

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Find the harmonic mean between $\frac{a}{(1 - a b)} and \frac{a}{(1 + a b)}$ .

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

The

AM

of two numbers exceeds their

GM

by

10

and

HM

by

16

. Find the numbers.

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

If

a_{i} > 0, i = 1, 2, 3, . . ., 50

and

a_{1} + a_{2} + a_{3} + . . . + a_{50} = 50

, then the minimum value of

\frac{1}{a_{1}} + \frac{1}{a_{2}} + \frac{1}{a_{3}} + . . . + \frac{1}{a_{50}}

is equal to.

EASY

Mathematics>Algebra>Sequence and Series>Harmonic Progression

The harmonic mean of

4, 8, 16

is

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Let

A, G, H

be

A . M ., G . M .

and

H . M .

of three positive real numbers

a, b, c

, respectively such that

G^{2} = A H,

then prove that

a, b, c

are terms of a

G . P .

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

If

p, q, r

in harmonic progression and

p & r

be different having same sign then the roots of the equation

p x^{2} + q x + r = 0

are

EASY

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Find the harmonic mean between $7 and 9$

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Harmonic mean of the reciprocal of even numbers from

12

to

190

is

EASY

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Cotyledons are also called-