Strange Fraction

Algebra Level pending

The value of $\pi$ is approximated using the following complex fraction:

$\pi \approx k_{1} + \frac{1}{ k_{2} + \frac{1}{ k_{3} + \frac{1}{ k_{4} + \frac{1}{ k_{5} } } } }$

There are, however, a couple of constraints to the constants $k_{1}, k_{2}, k_{3}, k_{4}, k_{5}$ :

$\sum^{n=5}_{n=1}k_{n} = 318$
All constants are positive integers.
The error between the actual value of pi and the above approximation is less than $10^{-9}$ .

If the value of this complex fraction is equal to $\frac{a}{b}$ , where $a$ and $b$ are relatively prime integers (that is, $\gcd(a, b) = 1$ ), what is $a + b$ ?

The answer is 137095.

2 solutions

Alexander McDowell
Mar 12, 2020

The complex fraction above is a segment of the continued fraction representation of $\pi$ :

$\pi \approx 3 + \frac{1}{7 + \frac{1}{15 + \frac{1}{1 + \frac{1}{292 + ...}}}}$

The derivation of the continued fraction representation of $\pi$ can be found at this stack exchange post .

Substituting $k_{1} = 3, k_{2} = 7, k_{3} = 15, k_{4} = 1, k_{5} = 292$ ( $\sum^{n=5}_{n=1}k_{n} = 318$ ) into the given complex fraction and simplifying into a single fraction yields $\frac{103993}{33102}$ . The numerator and denominator are relatively prime ( $\gcd(103993, 33102) = 1$ ) to each other and $| \pi - \frac{103993}{33102} | = 5.7789 * 10^{-10}$ , therefore the answer is $103993 + 33102 =137095$ .

Yuriy Kazakov
May 16, 2020

A001203

Strange Fraction

The answer is 137095.

2 solutions

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