General term of the series

Question

Given the series:

$$\sqrt c + \sqrt{c\sqrt c} + \sqrt{c\sqrt{c\sqrt c}} + \ldots$$

where $0 < c < 1$

What is the general term of this series?

Answer 1

$a_1 = \sqrt{c} = c^{1/2}$, $a_2 = \sqrt{c\sqrt{c}} = \sqrt{c^{3/2}} = c^{3/4}$, $a_3 = \sqrt{c\sqrt{c\sqrt{c}}} = \sqrt{c \times c^{3/4}} = c^{7/8}$. In general, $$a_{n+1} = \sqrt{c a_{n-1}}$$ with $a_0 = 1$. This gives us $a_n = c^{1-1/2^n}$

Answer 2

Each term can be written as $a_n = a_{n-1}c^{2^{-n}}$. Since $a_1 = c^{1/2}$, this gives $$a_n = c^{\;\sum\limits_{k=1}^n 2^{-k}} = c^{1-2^{-n}}$$