TITLE: How to evaluate $\sum\limits_{n \geq 0} \left(S_{n + 2} + S_{n + 1}\right)^2(-1)^n$, given the multivariable recurrence relation?
QUESTION [5 upvotes]: The given multivariable recurrence relation is that for every $n \geq 1$
$$S_{n + 1} = T_n - S_n$$
where $S_1 = \dfrac{3}{5}$ and $T_1 = 1$. Both $T_n$ and $S_n$ depend on the following condition
$$
\dfrac{T_n}{S_n} = \dfrac{T_{n + 1}}{S_{n + 1}} = \dfrac{T_{n + 2}}{S_{n + 2}} = \dots
$$
The goal is to evaluate
$$\sum\limits_{n \geq 0} \left(S_{n + 2} + S_{n + 1}\right)^2 (-1)^n$$
Since the change between $T_n$ and $T_{n + 1}$ is not constant, I believe that the way to approach this problem is to have all terms with consistent coefficient. However, I am not skillful enough to simplify the summation into a single variable.

REPLY [4 votes]: Notice that
$$\frac53=\frac{T_1}{S_1}=\frac{T_n}{S_n}$$
Thus, $T_n=\frac53S_n$.  Putting this in, we get
$$S_{n+1}=\frac23S_n$$
which is a geometric sequence.  The general form is then $S_n=\frac35\times\left(\frac23\right)^n$, so we have
$$\text{Sum}=\frac49\sum_{n\ge0}a^n$$
where $a=-\frac49$ is a very simple geometric series.