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 ...

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Standard XI

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

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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 $2$ .

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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