By bracketing the terms in pairs, or otherwise, find the sum of \(2n\) terms of the series \[1^2-3^2+5^2-7^2+\cdots.\]

Hence show that the sum of \(2n+1\) terms is \(8n^2+8n+1\).

Can we find an expression for a general term in the series?

What is the sign of the \(2k^{th}\) term?

Can we find expressions for the two terms in the \(k^{th}\) pair?

Can we factorise \(a^2-b^2\)?

Can we check our formula by substituting in a couple of values for \(n\)?