Question 19
Assume that $\{a_k\}_{k=1}^{\infty}$ is a sequence with the property that for any distinct positive integers $m,n,p,q$ with $m+n=p+q$, the following equality always holds:
$$\frac{a_m+a_n}{(a_m+1)(a_n+1)}=\frac{a_p+a_q}{(a_p+1)(a_q+1)}.$$
Given $a_1=0$ and $a_2=\dfrac12$, determine $\dfrac1{1-a_5}$.
(Hint: Consider $c_k=\dfrac1{a_k+1}-\dfrac12$ for all positive integer $k$.)
$$\frac{a_m+a_n}{(a_m+1)(a_n+1)}=\frac{a_p+a_q}{(a_p+1)(a_q+1)}.$$
Given $a_1=0$ and $a_2=\dfrac12$, determine $\dfrac1{1-a_5}$.
(Hint: Consider $c_k=\dfrac1{a_k+1}-\dfrac12$ for all positive integer $k$.)