Geometric sequence and its squares with given sums to infinity, and a logarithmic arithmetic sequence with a sum inequality

The terms \(u_1,u_2,u_3,\dots\) form a geometric sequence with common ratio \(r\), whose sum to infinity equals \(6\).

1. Express \(u_1\) in terms of \(r\).[1]

The squares \(u_1^{\,2},u_2^{\,2},u_3^{\,2},\dots\) also form a geometric sequence, with common ratio \(q\).

2. Show that \(q=r^2\).[2]

The squares have a sum to infinity of \(20\).

3. 1. Show that \(7r^2-9r+2=0\).
   2. Find \(r\).
   3. Hence find \(u_1\).[5]

A sequence \(w_1,w_2,w_3,\dots\) is defined by \(w_n=2k+\log_k u_n\), where \(k\in\mathbb{Z}\) and \(k>1\).

4. Show that \(w_1,w_2,w_3,\dots\) is an arithmetic sequence.[4]

The arithmetic sequence has \(W_{20}\) as the sum of its first \(20\) terms.

5. Find the largest value of \(k\) for which \(W_{20}<0\).[3]