Question 25
Let $p,q,r$ be positive numbers with $p-r=4q$, and $a_1,a_2,\ldots$ and $b_1,b_2,\ldots$ be two sequences defined by $a_1=p$, $b_1=q$ and for $n\ge2$,
$$a_n=pa_{n-1},\qquad b_n=qa_{n-1}+rb_{n-1}.$$
Find the value of $\displaystyle\lim_{n\to\infty}\frac{\sqrt{a_n^2+(3b_n)^2}}{b_n}$.
$$a_n=pa_{n-1},\qquad b_n=qa_{n-1}+rb_{n-1}.$$
Find the value of $\displaystyle\lim_{n\to\infty}\frac{\sqrt{a_n^2+(3b_n)^2}}{b_n}$.