N2026 P2 Q4

An arithmetic series has first term \(a\) and common difference \(d\), where \(a\neq 0\) and \(d\neq 0\). A geometric series has first term \(a\) and common ratio \(r\), where \(r>0\).

The 6th term of the arithmetic series is equal to the 5th term of the geometric series.

The 22nd term of the arithmetic series is equal to the 7th term of the geometric series.

1. Find the value of \(r\) and an expression for \(d\) in terms of \(a\).
   Describe the set of possible values of \(a\).[5]
2. The sum of the arithmetic series to \(37\) terms differs by \(12a\) from the sum of the geometric series to \(n\) terms. Find the value of \(n\).[4]