N2013 P1 Q7

A gardener is cutting off pieces of string from a long roll of string. The first piece he cuts off is \(128\text{ cm}\) long and each successive piece is \(\frac23\) as long as the preceding piece.

1. The length of the \(n\)th piece of string cut off is \(p\text{ cm}\). Show that \(\ln p=(An+B)\ln2+(Cn+D)\ln3\), for constants \(A\), \(B\), \(C\) and \(D\) to be determined.[3]
2. Show that the total length of string cut off can never be greater than \(384\text{ cm}\).[2]
3. How many pieces must be cut off before the total length cut off is greater than \(380\text{ cm}\)? You must show sufficient working to justify your answer.[4]