A Lucas Sum
Algebra
Level
5
Let \(p = \sqrt{1+\sqrt{1+\sqrt{1+\ldots}}}\).
The sum
\[\sum_{k=2}^{\infty} \frac{\lfloor p^{k} \rceil}{2^{k}}\]
can be expressed as \(\frac{a}{b}\) for \(a,b\) coprime, where \(\lfloor \cdot \rceil\) denotes the nearest integer function. Find \(a+b\).
by
Jake Lai