Continued fractions given π \pi

Calculus Level 5

1 6 + 3 2 6 + 5 2 6 + 7 2 6 + 9 2 6 + ⋱ \cfrac1{6 + \cfrac{3^2}{6 + \cfrac{5^2}{6 + \cfrac{7^2}{6 + \cfrac{9^2}{6 + _ \ddots}}}}}

The above represents the continued fractions of the number A π − B A \pi - B , where A A and B B are positive integers. Find A + B A+B .


The answer is 4.

Hamza A by Hamza A , 19, USA

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1 solution

Jack Lam
Jul 7, 2016

One of the continued fraction representations of π \pi are as follows:

π = 3 + 1 6 + 3 2 6 + 5 2 6 + 7 2 6 + 9 2 6 + ⋱ \pi = 3+\cfrac{1}{6+\cfrac{3^2}{6+\cfrac{5^2}{6+\cfrac{7^2}{6+\cfrac{9^2}{6+_\ddots}}}}}

From which the result follows immediately by subtracting 3 from both sides.

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