The seven numbers \(a, x_1, x_2, x_3, x_4, x_5, b\) are in arithmetic progression. Express \(x_2\) in terms of \(a\) and \(b\), and show that \(x_1+x_3+x_5=\dfrac{3}{2}(a+b)\).

Given also that the numbers \(a, x_2, b\) are in geometric progression, and that \(b\neq a\), express \(b\) in terms of \(a\).

A geometric series has common ratio \(r,\) where \(|r|<1.\) The sum of the first \(n\) terms of the series is \(S_n\) and the sum to infinity is \(S\). Express \(r\) in terms of \(S_n, S\) and \(n,\) and prove that the sum of the first \(2n\) terms of the series is \(\dfrac{S_n(2S-S_n)}{S}\).