How can i find the general term for this sequence?
How can I determine the general term of the sequence below:
$\sqrt2, \sqrt{2\sqrt2}, \sqrt{2\sqrt{2\sqrt2}},...$.
I've tried checking if it's some sort of geometric sequence, but it doesn't seem so.
Try writing every term of the sequence in expontial form with basis 2 and you will quickly see that the sequence is
\begin{equation} 2^{\frac{1}{2}},2^{\frac{1}{2}+\frac{1}{4}},2^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}},\ldots \end{equation}
Thus, $x_n = 2^{\sum_{i=1}^n 2^{-i}}$. The exponent is a geometric series that converges to $1$. Since exponentiation is continuous, the sequence converges to $2$.