Expanding the Idea!

Algebra Level 3

$\large\frac{1-\frac{1}{2^3}}{1+\frac{1}{2^3}}\times\frac{1-\frac{1}{3^3}}{1+\frac{1}{3^3}}\times\dots\dots\times \frac{1-\frac{1}{2018^3}}{1+\frac{1}{2018^3}}=\frac{2}{3}\left(1+\frac{1}{A\times B}\right)$

What is the minimum value of $A+B$ , where $A$ and $B$ are positive integers?

Keep in mind: We need $|A-B|$ to be minimized.

The answer is 4037.

1 solution

A Former Brilliant Member
Dec 21, 2018

Using induction on n, we get : (1-1/2^3)/(1+1/2^3) (1-1/3^3)/(1+1/3^3) ...(1-1/n^3)/(1+1/n^3)=2/3(1+1/n(n+1)) Therefore A=n and B=n+1. For n=2018, A+B=2018+2019=4037

Expanding the Idea!

The answer is 4037.

1 solution

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