Question 30
Show that if $a_{1} / b_{1}=a_{2} / b_{2}=a_{3} / b_{3}$ and $p_{1}, p_{2}, p_{3}$ are not all zero, then
$$
\left(\frac{a_{1}}{b_{1}}\right)^{n}=\frac{p_{1} a_{1}^{n}+p_{2} a_{2}^{n}+p_{3} a_{3}^{n}}{p_{1} b_{1}^{n}+p_{2} b_{2}^{n}+p_{3} b_{3}^{n}}
$$
for every positive integer $n$.
$$
\left(\frac{a_{1}}{b_{1}}\right)^{n}=\frac{p_{1} a_{1}^{n}+p_{2} a_{2}^{n}+p_{3} a_{3}^{n}}{p_{1} b_{1}^{n}+p_{2} b_{2}^{n}+p_{3} b_{3}^{n}}
$$
for every positive integer $n$.