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

MEDIUM

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Let the harmonic mean and geometric mean of two positive number be in the ratio $4 : 5$ . then the ratio of two number is $λ (λ < 1),$ find $11 λ$ .

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Important Questions on Sequences and Series

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

Let

a, b

and

c

be the

7^{t h}, 11^{t h}

and

13^{t h}

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>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

HARD

Mathematics>Algebra>Sequences and Series>Harmonic 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

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic 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:

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

The sum of first

20

terms of the sequence

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

is :

HARD

Mathematics>Algebra>Sequences and Series>Harmonic 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

HARD

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

Let

x, y, z

be three non-negative integers such that

x + y + z = 10

. The maximum possible value of

x y z + x y + y z + z x

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

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

Three positive numbers form an increasing

G . P .

If the middle term in this

G . P .

is doubled, the new numbers are in

A . P .

Then the common ratio of the

G . P .

is :

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

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

Suppose

p, q, r

are real numbers such that

q = p (4 - p), r = q (4 - q), p = r (4 - r) .

The maximum possible value of

p + q + r

HARD

Mathematics>Algebra>Sequences and Series>Harmonic 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.)

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

The quotient when

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

is divided by

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

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

Let

f : x \to y

be such that

f (1) = 2

and

f (x + y) = f (x) f (y)

for all natural numbers

x

and

y

.

\sum_{k = 1}^{n} f (a + k) = 16 (2^{n} - 1)

then

a

is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Harmonic Progressions

The sum of the first

20

terms of the series

1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \dots

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

Let the harmonic mean and geometric mean of two positive number be in the ratio 4: 5. then the ratio of two number is λ(λ<1), find 11λ.

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

Let the harmonic mean and geometric mean of two positive number be in the ratio $4 : 5$ . then the ratio of two number is $λ (λ < 1),$ find $11 λ$ .