Question 26
The sequence $\{a_{n}\}_{n=1}^{\infty}$ is defined by $a_{1}=0$ and $a_{n+1}= a_{n}+4 n+3$, $n \geq 1$.
a) Express $a_{n}$ as a function of $n$.
b) Find the limit
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{a_{n}}+\sqrt{a_{4 n}}+\sqrt{a_{4^{2} n}}+\cdots+\sqrt{a_{4^{10} n}}}{\sqrt{a_{n}}+\sqrt{a_{2 n}}+\sqrt{a_{2^{2} n}}+\cdots+\sqrt{a_{2^{10} n}}}
$$
a) Express $a_{n}$ as a function of $n$.
b) Find the limit
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{a_{n}}+\sqrt{a_{4 n}}+\sqrt{a_{4^{2} n}}+\cdots+\sqrt{a_{4^{10} n}}}{\sqrt{a_{n}}+\sqrt{a_{2 n}}+\sqrt{a_{2^{2} n}}+\cdots+\sqrt{a_{2^{10} n}}}
$$