Problem (Algebra):
In an arithmetic sequence with 5 terms, the sum of the squares of the first 3 terms equals the sum of the squares of the last 2 terms. If the first term is 5 , determine all possible values of the fifth term.

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3,5,7,9,11 is an arithmetic sequence with five terms.)

Solution:
Let the common difference in this arithmetic sequence be $d$.

Since the first term in the sequence is 5 , then the 5 terms are $5,5+d, 5+2 d, 5+3 d, 5+4 d$.

From the given information, $5^{2}+(5+d)^{2}+(5+2 d)^{2}=(5+3 d)^{2}+(5+4 d)^{2}$.

Manipulating, we obtain the following equivalent equations:

$$
\begin{aligned}
5^{2}+(5+d)^{2}+(5+2 d)^{2} & =(5+3 d)^{2}+(5+4 d)^{2} \\
25+\left(25+10 d+d^{2}\right)+\left(25+20 d+4 d^{2}\right) & =\left(25+30 d+9 d^{2}\right)+\left(25+40 d+16 d^{2}\right) \\
75+30 d+5 d^{2} & =50+70 d+25 d^{2} \\
0 & =20 d^{2}+40 d-25 \\
0 & =4 d^{2}+8 d-5 \\
0 & =(2 d+5)(2 d-1)
\end{aligned}
$$

Therefore, $d=-\frac{5}{2}$ or $d=\frac{1}{2}$.

These give possible fifth terms of $5+4 d=5+4\left(-\frac{5}{2}\right)=-5$ and $5+4 d=5+4\left(\frac{1}{2}\right)=7$.

(We note that, for these two values of $d$, the sequences are $5, \frac{5}{2}, 0,-\frac{5}{2},-5$ and $5, \frac{11}{2}, 6, \frac{13}{2}, 7$.)

Final answer: -5,7