Multiplying Successive Square Roots

Algebra Level 1

A 1 = 3 A 2 = A 1 A 3 = A 2 ⋮ A n + 1 = A n \begin{aligned} A_1 & = \sqrt{3} \\ A_2 & = \sqrt{A_1} \\ A_3 & = \sqrt{A_2} \\ & \vdots \\ A_{n+1} &= \sqrt{A_n} \end{aligned}

What is the value of the infinite product A 1 × A 2 × A 3 × ⋯ ? A_1 \times A_2 \times A_3\times \cdots \, ?


The answer is 3.

Steven Chase by Steven Chase , 37, USA

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1 solution

Relevant wiki: Geometric Progression Sum

Let A 1 = a 1 2 A_1=a^\frac 12 , then A n = a 1 2 n A_n = a^\frac 1{2^n} . We need to find:

P = ∏ n = 1 ∞ A k = ∏ n = 1 ∞ a 1 2 n = a ∑ n = 1 ∞ 1 2 n = a 1 2 ( 1 1 − 1 2 ) = a 1 \begin{aligned} P & = \prod_{n=1}^\infty A_k = \prod_{n=1}^\infty a^\frac 1{2^n} = \large a^{\sum_{n=1}^\infty \frac 1{2^n}} = a^{\frac 12 \left(\frac 1{1-\frac 12}\right)} = a^1 \end{aligned}

For a = 3 a=3 , the answer is 3 \boxed{3} .

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