If -ni-1ai -1 and -ni-11-ai - xn- where ai- i-1- 2- - n is positive real numbers- then the value of x is

Using A.M ≥ G.M 1+a_1 ≥ 2√(a_1) 1+a_2 ≥ nbsp; 2√(a_2) . . . 1+a_n ≥ 2√(a_n) From above, we get nbsp; ⇒ (1+a_1) (1+a_2) … (1+a_n) ≥ 2^n(a_1 a_2 a_3 … a_n ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Arithmetic Mean

If ∏ni=1ai =1...

Question

If $n \prod i = 1 a_{i} = 1$ and $n \prod i = 1 (1 + a_{i}) \geq x^{n}$ , where $a_{i}, i = 1, 2, \dots, n$ is positive real numbers, then the value of $x$ is

Open in App

Solution

Using
$A . M \geq G . M$
$1 + a_{1} \geq 2 \sqrt{a_{1}}$
$1 + a_{2} \geq 2 \sqrt{a_{2}} . . . 1 + a_{n} \geq 2 \sqrt{a_{n}}$
From above, we get
$\Rightarrow (1 + a_{1}) (1 + a_{2}) \dots (1 + a_{n}) \geq 2^{n} (a_{1} a_{2} a_{3} \dots a_{n})^{1 / 2} \Rightarrow n \prod i = 1 (1 + a_{i}) \geq 2^{n} {(n \prod i = 1 a_{i})}^{1 / 2} ∴ n \prod i = 1 (1 + a_{i}) \geq 2^{n} (∵ n \prod i = 1 a_{i} = 1)$

Hence, the value of $x$ is $2$ .

Suggest Corrections

Similar questions

Q. If

n \prod i = 1 a_{i} = 1

and

n \prod i = 1 (1 + a_{i}) \geq x^{n}

, where

a_{i}, i = 1, 2, \dots, n

is positive real numbers, then the value of

x

Q. If

n \prod i = 1 a_{i} = 1

and

n \prod i = 1 (1 + a_{i}) \geq x^{n}

, where

a_{i}, i = 1, 2, \dots, n

is positive real numbers, then the value of

x

Q. If

\sum_{i = 1}^{n}_{i} = 0

where

∣ ∣_{i} ∣ ∣ = 1

for all

i

then the value of

\sum_{1 \leq i < j \leq n} \sum_{i}_{j}

Q. If

n \sum i = 1 {\to a}_{i} = 0

where

∣ ∣ {\to a}_{i} ∣ ∣ = 1, \forall i,

then the value of

\sum 1 \leq i < j \leq n \sum {\to a}_{i} . {\to a}_{j}

is equal to

A_{i} = \frac{x - a_{i}}{| x - a_{i} |}, i = 1, 2, . . . . ., n,

and

a_{1} < a_{2} < a_{3} < . . . . . < a_{n} .

1 \leq m \leq n, m \in N,

then the value of

R = lim x \to a_{m}^{+} (A_{1} A_{2} . . . . A_{n})

Join BYJU'S Learning Program

If n∏i=1ai=1 and n∏i=1(1+ai)≥xn, where ai,i=1,2,…,n is positive real numbers, then the value of x is

Using A.M≥G.M 1+a1≥2√a1 1+a2≥2√a2...1+an≥2√an From above, we get ⇒(1+a1)(1+a2)…(1+an)≥2n(a1a2a3…an)1/2⇒n∏i=1(1+ai)≥2n(n∏i=1ai)1/2∴n∏i=1(1+ai)≥2n(∵n∏i=1ai=1) Hence, the value of x is 2.

If $n \prod i = 1 a_{i} = 1$ and $n \prod i = 1 (1 + a_{i}) \geq x^{n}$ , where $a_{i}, i = 1, 2, \dots, n$ is positive real numbers, then the value of $x$ is