(a) Use standard results from the list of formulae (MF19) to show that

\(\sum_{r=1}^{N} r(r+1)(3r+4) = \frac{1}{12}N(N+1)(N+2)(9N+19).\)

(b) Express \(\frac{3r+4}{r(r+1)}\) in partial fractions and hence use the method of differences to find

\(\sum_{r=1}^{N} \frac{3r+4}{r(r+1)} \left( \frac{1}{4} \right)^{r+1}\)

in terms of \(N\).

(c) Deduce the value of

\(\sum_{r=1}^{\infty} \frac{3r+4}{r(r+1)} \left( \frac{1}{4} \right)^{r+1}.\)