91Ó°ÊÓ

(a) Assuming the formula for \(\sum_{r=1}^{n} r^{2}\), write down (i) the sum of the squares of the first \(2 n\) positive integers, (ii) the sum of the squares of the first \(n\) even integers. Hence find \(1^{2}+3^{2}+5^{2}+\ldots+(2 n-1)^{2}\) (b) If \(S_{n}=\sum_{r=0}^{n} a^{r}\left(1+a+a^{2}+\ldots+a^{r}\right), \quad(|a| \neq 1)\) by considering \((1-a) S_{n}\), show that $$ S_{n}=\frac{1-a^{2 n+2}}{\left(1-a^{2}\right)(1-a)}-\frac{a^{n+1}\left(1+a^{n+1}\right)}{(1-a)^{2}} $$ State the set of values of \(a\) for which \(S_{n}\) approaches a limit as \(n \rightarrow \infty\) and find the sum to infinity of the series for these values of \(a\). (U of L)

The positive integers are bracketed as follows: $$ (1),(2,3),(4,5,6), \ldots $$ where there are \(r\) integers in the \(r\) th bracket. Find expressions for the first and last integers in the \(r\) th bracket. Find the sum of all the integers in the first 20 brackets. Prove that the sum of the integers in the \(r\) th bracket is \(\frac{1}{2}\left(r^{2}+1\right)\).

(a) Prove that \(\sum_{r=1}^{n} \frac{1}{r(r+1)}=\frac{n}{n+1}\) (b) Sum the series \(1+x+x^{2}+\ldots+x^{n}\) for \(x \neq 1\) By differentiation with respect to \(x\), or otherwise, find the value of $$ 1+2 x+3 x^{2}+\ldots+n x^{n-1} $$ and deduce the value of $$ 1.2+2 \cdot 2^{2}+3 \cdot 2^{3}+\ldots+n \cdot 2^{n} $$ (U of \(\mathrm{L})\)

Prove that $$ 1^{2}+2^{2}+3^{2}+\ldots+n^{2}=\frac{1}{6} n(n+1)(2 n+1) $$ Show that $$ \begin{aligned} &a^{2}+(a+d)^{2}+(a+2 d)^{2}+\ldots+(a+n d)^{2} \\ &\quad=\frac{1}{6}(n+1)\left[6 a(a+n d)+d^{2} n(2 n+1)\right] \end{aligned} $$ Hence, or otherwise, prove that $$ \begin{aligned} &2^{2}+4^{2}+6^{2}+\ldots+l^{2}=\frac{1}{6} l(l+1)(l+2) \quad(l \text { even }) \\ &\text { and } \quad 1^{2}+3^{2}+5^{2}+\ldots+l^{2}=\frac{1}{6} l(l+1)(l+2) \quad(l \text { odd }) \quad(\mathrm{JMB}) \end{aligned} $$