N2009 P1 Q8

Two musical instruments, \(A\) and \(B\), consist of metal bars of decreasing lengths.

1. The first bar of instrument \(A\) has length \(20\) cm and the lengths of the bars form a geometric progression. The \(25\)th bar has length \(5\) cm. Show that the total length of all the bars must be less than \(357\) cm, no matter how many bars there are.[4]

Instrument \(B\) consists of only \(25\) bars which are identical to the first \(25\) bars of instrument \(A\).

2. Find the total length, \(L\) cm, of all the bars of instrument \(B\) and the length of the \(13\) th bar.[3]
3. Unfortunately the manufacturer misunderstands the instructions and constructs instrument \(B\) wrongly, so that the lengths of the bars are in arithmetic progression with common difference \(d\) cm. If the total length of the \(25\) bars is still \(L\) cm and the length of the \(25\)th bar is still \(5\) cm, find the value of \(d\) and the length of the longest bar.[4]