Question 1

For each positive integer $n$, let $s_n$ be given by the following sum:

$$s_n = 1^2 + 3^2 + 5^2 + \dots + (2n -1)^2, \hspace{3pt} \textrm{where there are \hspace{2pt}} n\hspace{2pt} \textrm{ terms in the sum.}$$

Prove that, for each positive integer $n$, we have:

$$s_n = \frac{1}{3} n(2n - 1)(2n +1).$$

Also determine, with proof, the following limits, or explain why the limit in question does not exist:

• $\mathcal{A} = \displaystyle{ \lim_{n\rightarrow \infty} \left(\frac{s_n}{n(n+1)(n + 2)}\right)}$
  
  
  
  
• $\mathcal{B} = \displaystyle{\lim_{n\rightarrow \infty}\left( \frac{s_{2n+1} - s_{2n}}{4n^2 -1}\right)}$