An algebra problem by Sabhrant Sachan

Algebra Level 3

If $a_i > 0$ for $i=1,2,3,\ldots, n$ and $a_1 a_2 a_3\cdots a_n = 1$ , find the minimum value of $(1+a_1)(1+a_2 )\cdots (1+a_n)$ .

2

2^{n/2}

2^{n}

1

2^{2n}

1 solution

Mohammed Imran
Apr 16, 2020

Applying AM-GM inequality, we have $(a_{1}+1)(a_{2}+1)...(a_{n}+1) \geq 2 \sqrt {a_{1}} \times 2\sqrt{a_{2}}...2\sqrt{a+{n}}=2^n[1]=\boxed{2^n}$