Find a formula for a sequence $\{\sqrt{3},\sqrt{3\sqrt{3}},\sqrt{3\sqrt{3\sqrt{3}}},...\}$
By Sarah Scott •
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?
$\endgroup$ 42 Answers
$\begingroup$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.
$\endgroup$ $\begingroup$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 $a_n = 3^{1-(\frac{1}{2})^n}$
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