The two sequences \(a_0\), \(a_1\), \(a_2\), \(\ldots\)
and
\(b_0\), \(b_1\), \(b_2\), \(\ldots\)
have general terms
\[
a_n = \lambda^n +\mu^n
\text { \ \ \ and \ \ \ }
b_n = \lambda^n - \mu^n\,,
\]
respectively, where \(\lambda = 1+\sqrt2\) and \(\mu= 1-\sqrt2\,\).
- Show that $\displaystyle \sum_{r=0}^nb_r = -\sqrt2 + \frac
1 {\sqrt2} \,a_{\low n+1}\,$,
and give a corresponding result for
\(\displaystyle \sum_{r=0}^na_r\,\).
- Show that, if \(n\) is odd,
$$\sum_{m=0}^{2n}\left( \sum_{r=0}^m a_{\low r}\right)
= \tfrac12 b_{n+1}^2\,,$$
and give a corresponding result when \(n\)
is even.
- Show that, if \(n\) is even,
$$\left(\sum_{r=0}^na_r\right)^{\!2}
-\sum_{r=0}^n a_{\low 2r+1} =2\,,$$
and give a corresponding result when
\(n\) is odd.