Calculus - Inadequate Information #1

$a_n$ and $b_n$ never converges, so it is natural to try and find the limiting value of $\dfrac{b_n}{a_n}.$

Divide the second equation by the first.

$\dfrac{b_{n+1}}{a_{n+1}}=\dfrac{a_n+b_n}{a_n+2b_n}.$ Divide the numerator and denominator by $a_n$ on the right side.

$\dfrac{b_{n+1}}{a_{n+1}}=\dfrac{1+\dfrac{b_n}{a_n}}{1+2\cdot\dfrac{b_n}{a_n}}.$ Set $\dfrac{b_n}{a_n}=c_n.$

$c_{n+1}=\dfrac{1+c_n}{1+2c_n}.$ It is clear that $c_n>0$ for every natural number $n,$ and therefore $\{c_n\}$ converges.

Let $\displaystyle \lim_{n\to\infty} c_n=\alpha$ and we see that

$\alpha=\dfrac{1+\alpha}{1+2\alpha},$ or $2\alpha^2=1.$

Since $\alpha>0,$ we see that $\displaystyle \lim_{n\to\infty}\dfrac{b_n}{a_n}=\lim_{n\to\infty}c_n=\alpha=\dfrac{\sqrt{2}}{2}.$

Therefore $N+Q+D=1+2+2=\boxed{5}.$