Find a formula for a sequence $\{\sqrt{3},\sqrt{3\sqrt{3}},\sqrt{3\sqrt{3\sqrt{3}}},...\}$

Question

I'm trying to find a formula for the following sequence:

$\{\sqrt{3},\sqrt{3\sqrt{3}},\sqrt{3\sqrt{3\sqrt{3}}},...\}$

I thought of solving it recursively and I got this formula:

$a_{n}=\sqrt{3*a_{n-1}}$

$a_{0}=1$

Is there a better and non-recursive formula for the given sequence?

Answer 1

If we start with $a_0$, what about $a_n=3^{\left(1-\frac{1}{2^{n+1}}\right)}$? Note that the terms are $3^{1/2}$, $3^{3/4}$, $3^{7/8}$, and so on.

Answer 2

Let $b_n = \ln( a_n )$, then according to $a_n = \sqrt{3a_{n-1}}$, we have:

$b_n = \frac{1}{2}[ \ln(3)+b_{n-1} ]$ with $b_0=0$

which is a classical problem. We can easily find its solution: $b_n = \ln(3)[ 1 - (\frac{1}{2})^n]$. It is trivial to convert $b_n$ to $a_n = 3^{1-(\frac{1}{2})^n}$