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…

Let $\displaystyle \left\{a_k\right\}$ and $\displaystyle \left\{b_k\right\}, k \in \mathbb{N}$, be two G.P.s with common ratios $\displaystyle \mathrm{r}_1$ and $\displaystyle \mathrm{r}_2$ respectively such that $\displaystyle a_1=\mathrm{b}_1=4$ and $\displaystyle \mathrm{r}_1<\mathrm{r}_2$. Let $\displaystyle \mathrm{c}_k=a_k+\mathrm{b}_k, k \in \mathbb{N}$. If $\displaystyle \mathrm{c}_2=5$ and $\displaystyle \mathrm{c}_3=\frac{13}{4}$ then $\displaystyle \sum_{k=1}^{\infty} \mathrm{c}_k-\left(12 a_6+8 b_4\right)$ is equal to $\displaystyle \_\_\_\_$.