It is given that \(\sum_{r=1}^{n} \frac{2}{r(r+1)(r+2)}=\frac{1}{2}-\frac{1}{(n+1)(n+2)}\).
State the sum to infinity of the series.[1]
Find the sum of the first \(n\) terms of the series \(\frac{2}{17\times18\times19}+\frac{2}{18\times19\times20}+\frac{2}{19\times20\times21}+\cdots.\)[2]
Find \(\sum_{r=2}^{n} \frac{1}{r(r^{2}-1)}\).[3]
The sequence \(u_1, u_2, u_3, \ldots\) is defined by \(u_1=3\), \(u_{n+1}=2-\frac{4}{u_n}\), \(n\geq 1\).
Find the values of \(u_2\), \(u_3\), \(u_4\) and \(u_{100}\).[2]
Hence find \(\sum_{r=1}^{3n-2} u_r\) in terms of \(n\), simplifying your answer.[2]
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