Solutions - 0150-05

(a)

$$\begin{array}{c}{a}_3=S_3-S_2\gg \gg (5-\frac{2}{3^2})-(5-\frac{2}{2^2})\\ \\ \gg \gg -\frac{2}{9}+\frac{2}{4}\gg \gg \frac{5}{18}\end{array}$$

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(b)

$$\begin{array}{c}{a}_n=S_n-S_{n-1}\gg \gg (5-\frac{2}{n^2})-(5-\frac{2}{{(n-1)}^2})\\ \\ \gg \gg \frac{2}{{(n-1)}^2}-\frac{2}{n^2}\\ \\ \gg \gg \frac{2n^2-2{(n-1)}^2}{n^2{(n-1)}^2}\\ \\ \gg \gg \frac{2n^2-2(n^2-2n+1)}{n^2{(n-1)}^2}\\ \\ \gg \gg \frac{4n-2}{n^2{(n-1)}^2}\end{array}$$

When $n=1$, this formula is undefined, because $S_0$ is undefined. But we know that:

$$a_1=S_1\gg \gg 5-\frac{2}{1^2}\gg \gg 3$$

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(c)

Simply take the limit of $S_N$ as $N\to \infty$:

$$\begin{array}{c}{S}_N\stackrel{n\to \infty }{⟶}S\\ \\ 5-\frac{2}{N^2}\stackrel{n\to \infty }{⟶}5\end{array}$$