Question

Let ${a_k}$ and ${b_k}$, $𝑘∈N$, be two G.P.s with common ratios $r_1$ and $r_2$ respectively such that $a_1= b_1=4$ and $r_1<r_2$. Let $c_k=a_k+b_k$, $k∈N$. If $c_2=5$ and $c_3=\frac {13}{4}$, then $\displaystyle\sum_{k=1}^{∞ }c_k−(12a_6+8b_4)$ is equal to _____ .

Answer 1

The answer is 9.