Given that \(\sum\limits_{r=2}^{n}{\frac{2}{(r+3)(r+5)}}=\frac{11}{30}-\frac{2n+9}{\left( n+4 \

Question

Given that $\sum\limits_{r=2}^{n}{\frac{2}{(r+3)(r+5)}}=\frac{11}{30}-\frac{2n+9}{\left( n+4 \right)\left( n+5 \right)}$, find $\sum\limits_{r=5}^{n+4}{\frac{2}{r(r+2)}}$ $\sum\limits_{r=4}^{n+4}{\frac{2}{r(r+2)}}$

Accepted Answer

(i) $\frac{11}{30}-\frac{2n+11}{(n+5)(n+6)}$ (ii) $\frac{9}{20}-\frac{2n+11}{(n+5)(n+6)}$

Sigma Notation By Substitution Q2

Sigma Notation By Substitution Q2

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