2021 DHS P2 Q2

By expressing \(\frac{1}{{{2}^{k+1}}-3}-\frac{1}{{{2}^{k}}-3}\) as a single fraction, find in terms of \(N\), an expression for \(\sum\limits_{k=1}^{N}{\frac{{{2}^{k+1}}}{\left( {{2}^{k+1}}-3 \right)\left( {{2}^{k}}-3 \right)}}\).[4]

Use your result in (i), or otherwise, to evaluate

\(\frac{16}{\left( 61 \right)\left( 29 \right)}+\frac{32}{\left( 125 \right)\left( 61 \right)}+\frac{64}{\left( 253 \right)\left( 125 \right)}+...+\frac{{{2}^{k-1}}}{\left( {{2}^{k+1}}-3 \right)\left( {{2}^{k}}-3 \right)}+...,\)

leaving your answer as a fraction in its lowest terms.[3]