Problem (Algebra):
Determine all possible values of $r$ such that the three term geometric sequence 4, $4 r, 4 r^{2}$ is also an arithmetic sequence.

(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.)

Solution:
Since the sequence $4,4 r, 4 r^{2}$ is also arithmetic, then the difference between $4 r^{2}$ and $4 r$ equals the difference between $4 r$ and 4 , or

$$
\begin{aligned}
4 r^{2}-4 r & =4 r-4 \\
4 r^{2}-8 r+4 & =0 \\
r^{2}-2 r+1 & =0 \\
(r-1)^{2} & =0
\end{aligned}
$$

Therefore, the only value of $r$ is $r=1$.

Since the sequence $4,4 r, 4 r^{2}$ is also arithmetic, then we can write $4 r=4+d$ and $4 r^{2}=4+2 d$ for some real number $d$. (Here, $d$ is the common difference in this arithmetic sequence.)

Then $d=4 r-4$ and $2 d=4 r^{2}-4$ or $d=2 r^{2}-2$.

Therefore, equating the two expressions for $d$, we obtain $2 r^{2}-2=4 r-4$ or $2 r^{2}-4 r+2=0$ or $r^{2}-2 r+1=0$ or $(r-1)^{2}=0$.

Therefore, the only value of $r$ is $r=1$.

Final answer: 1