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The product of $n$ positive numbers is $1$ . Then the minimum value of their sum is______.

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Important Questions on Sequence and Series

HARD

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

Let

m

be the minimum possible value of

\log_{3} (3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}})

, where

y_{1}, y_{2}, y_{3}

are real numbers for which

y_{1} + y_{2} + y_{3} = 9

. Let

M

be the maximum possible value of

(\log_{3} x_{1} + \log_{3} x_{2} + \log_{3} x_{3})

, where

x_{1}, x_{2}, x_{3}

are positive real numbers for which

x_{1} + x_{2} + x_{3} = 9

. Then the value of

\log_{2} (m^{3}) + \log_{3} (M^{2})

HARD

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

Let

x, y, z

be positive real numbers such that

x + y + z = 12

and

x^{3} y^{4} z^{5} = (0 .1) {(600)}^{3} .

Then

x^{3} + y^{3} + z^{3}

is equal to

MEDIUM

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

The minimum value of

2^{\sin x} + 2^{\cos x}

is :

EASY

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

Let

I (n) = n^{n}, J (n) = 1.3.5 \dots \dots . . (2 n - 1)

for all

(n > 1), n \in N,

then

HARD

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

Let

x, y, z

be positive reals. Which of the following implies

x = y = z ?

x^{3} + y^{3} + z^{3} = 3 x y z

II.

x^{3} + y^{2} z + y z^{2} = 3 x y z

II.

x^{3} + y^{2} z + z^{2} x = 3 x y z

IV.

{(x + y + z)}^{3} = 27 x y z

EASY

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

a

and

b

are arbitrary positive real numbers then the least possible value of

\frac{6 a}{5 b} + \frac{10 b}{3 a}

MEDIUM

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

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)

MEDIUM

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

The minimum value of

f (x) = a^{a^{x}} + a^{1 - a^{x}}

, where

a, x \in R

and

a > 0

, is equal to:

MEDIUM

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

Let $n \geq 4$ be a positive integer and let $l_{1}, l_{2}, \dots, l_{n}$ be the lengths of the sides of arbitrary $n$ sided non-degenerate polygon $P$ . Suppose

$\frac{l_{1}}{l_{2}} + \frac{l_{2}}{l_{3}} + \dots + \frac{l_{n - 1}}{l_{n}} + \frac{l_{n}}{l_{1}} = n$

Consider the following statements:

$I$ . The lengths of the sides of $P$ are equal.
$II$ . The angles of $P$ are equal.
$III$ . $P$ is a regular polygon if it is cyclic. Then,

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Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

If the arithmetic mean of two numbers $a$ and $b, a > b > 0$ , is five times their geometric mean, then $\frac{a + b}{a - b}$ is equal to:

EASY

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

The number of three digit numbers

\bar{a b c}

such that the arithmetic mean of

b & c

and the square of their geometric mean are equal is

HARD

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

The number of triples

(x, y, z)

of real numbers satisfying the equation

x^{4} + y^{4} + z^{4} + 1 = 4 x y z

MEDIUM

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

The number of different possible values for the sum

x + y + z,

where

x, y, z

are real number such that

x^{4} + 4 y^{4} + 16 z^{4} + 64 = 32 x y z

EASY

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

x = \log_{5} 3 + \log_{7} 5 + \log_{9} 7

, then

EASY

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

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-

HARD

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

a, b, c

are positive numbers such that

a + b + c = 1

, then the minimum value of

\frac{1}{a b} + \frac{1}{b c} + \frac{1}{c a}

MEDIUM

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

Through the point

p (α, β)

, where

α β > 0

, the straight line

\frac{x}{a} + \frac{y}{b} = 1

is drawn so as to form with the coordinate axes a triangle of area

S

. If

a b > 0

, then the least value of

S

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Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

If three positive numbers

a

b

and

c

are in A.P. such that

a b c = 8

, then the minimum possible value of

b

is:

HARD

Mathematics>Algebra>Sequence and Series>A.M., G.M., H.M. and Their Relations

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>Sequence and Series>A.M., G.M., H.M. and Their Relations

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

The product of n positive numbers is 1. Then the minimum value of their sum is______.

Important Questions on Sequence and Series

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The product of $n$ positive numbers is $1$ . Then the minimum value of their sum is______.