2026 May TZC P2 Q9

The geometric sequence \(a_1, a_2, a_3, \dots\) has common ratio \(r\).
The sum to infinity \(A_{\infty}\), of this sequence is \(5\).

1. Find an expression for \(a_1\) in terms of \(r\).

   [1]

A second geometric sequence \(a_1^2, a_2^2, a_3^2, \dots\) has common ratio \(q\).

2. Show that \(q = r^2\).

   [2]

The sum to infinity, \(B_{\infty}\), of this second sequence is \(16\).

3. 1. Show that \(41r^2 - 50r + 9 = 0\).

   2. Find the value of \(r\).

   3. Hence, find the value of \(a_1\).

      [5]

A third sequence has its \(n\)th term given by \(c_n = \log_{k}{(a_n)} + k\), where \(k \in \mathbb{Z}^+, k > 1\).

4. Show that this third sequence is arithmetic.

   [4]

Let \(C_{20}\) denote the sum of the first \(20\) terms of the third sequence.

5. Find the maximum value of \(k\) such that \(C_{20} <0 \)

   [3]