Power of 2

Number Theory Level pending

True or False?

If x 1 , x 2 , … , x n x_1, x_2 , \ldots , x_n are positive numbers satisfying ( 1 + x 1 ) ( 1 + x 2 ) ( 1 + x 3 ) ⋯ ( 1 + x n ) ≥ 2 n , (1+x_1)(1+x_2)(1+x_3)\cdots(1+x_n) \geq 2^n , then x 1 x 2 x 3 ⋯ x n \sqrt{x_1x_2x_3 \cdots x_n} must be equal to 2.

True False

Ossama Ismail by Ossama Ismail , 63, Egypt

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1 solution

Ossama Ismail
Mar 28, 2017

Given that ( 1 + x 1 ) ( 1 + x 2 ) ( 1 + x 3 ) ⋯ ( 1 + x n ) ≥ 2 n , where x i > 0 , for i = 1 , 2 , ⋯ , n (1+x_1)(1+x_2)(1+x_3)\cdots(1+x_n) \geq 2^n , \text{where} \ x_i > 0 , \ \text{for} \ \ i = 1,2,\cdots, n

dividing the above equation by 2 n 2^n we got ( 1 + x 1 ) 2 ( 1 + x 2 ) 2 ( 1 + x 3 ) 2 ⋯ ( 1 + x n ) 2 ≥ 1 \dfrac{(1+x_1)}{2}\dfrac{(1+x_2)}{2}\dfrac{(1+x_3)}{2}\cdots \dfrac{(1+x_n)}{2} \geq 1

and

( 1 + x 1 ) 2 ( 1 + x 2 ) 2 ( 1 + x 3 ) 2 ⋯ ( 1 + x n ) 2 ≥ x 1 x 2 ⋯ x n = x 1 x 2 x 3 ⋯ x n = 1 \dfrac{(1+x_1)}{2}\dfrac{(1+x_2)}{2}\dfrac{(1+x_3)}{2} \cdots \dfrac{(1+x_n)}{2} \geq \sqrt{x_1}\sqrt{x_2} \cdots \sqrt{x_n} = \sqrt{x_1x_2x_3 \cdots x_n} = 1

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