3. An arithmetic sequence is a sequence where each term after the first is the sum of the previous term plus a constant value. For example, $3,7,11,15, \ldots$ is an arithmetic sequence.
$S$ is a sequence which has the following properties:
- The first term of $S$ is positive.
- The first three terms of $S$ form an arithmetic sequence.
- If a square is constructed with area equal to a term in $S$, then the perimeter of that square is the next term in $S$.
Determine all possible values for the third term of $S$.
Solution: Let the first term be $a^{4}$. By the third condition, the second term must be $4 a^{2}$ and the third term must be $8 a$. In an arithmetic sequence, the first term added to the third is twice the second term. So in our sequence we get:
$$
\begin{aligned}
a^{4}+8 a & =8 a^{2} \\
a^{4}-8 a^{2}+8 a & =0 \\
a(a-2)\left(a^{2}+2 a-4\right) & =0
\end{aligned}
$$

This gives solutions of $a=0,2,-1 \pm \sqrt{5}$. Since each of our terms represents the area or perimeter of a square, we need $a>0$. Thus, $a=2$ and $a=\sqrt{5}-1$, which gives 16 and $8 \sqrt{5}-8$ as the possible values for the third term.