CAIE A-Level Further Math AS 1.3.2 Method of Differences Questions

$\frac{1}{r^{3}}-\frac{1}{(r+1)^{3}}=\frac{(r+1)^{3}-r^{3}}{r^{3}(r+1)^{3}}=\frac{r^{3}+3 r^{2}+3 r+1-r^{3}}{r^{3}(r+1)^{3}}=\frac{3 r^{2}+3 r+1}{r^{3}(r+1)^{3}}$
Puts over a common denominator and
expands, AG.

$$\begin{aligned} \sum_{r=1}^{n} \frac{3 r^{2}+3 r+1}{r^{3}(r+1)^{3}}=\sum_{r=1}^{n}\left(\frac{1}{r^{3}}-\frac{1}{(r+1)^{3}}\right) =1-\frac{1}{2^{3}}+\frac{1}{2^{3}}-\frac{1}{3^{3}}+\ldots+\frac{1}{n^{3}}-\frac{1}{(n+1)^{3}} \end{aligned}$$
Shows three complete terms, including
last.

$1-\frac{1}{(n+1)^{3}}$
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