$ f(x)=a^x \forall x, y \in R $ $a_r=a_1+(r-1) d .\{$ Let $d$ is common difference of A. P) $ \begin{aligned} & \sum_{i=1}^{50} f\left(a_i\right)=\sum_{i=1}^{50} a^{\alpha_1+(i-1) d} \\ & =a^{\alpha_1-d} \sum_{i=1}^{50} a^{i d}=a^{\alpha_1-d}=\frac{a^d\left(1-\left(a^d\right)^{50}\right)}{1-a^d} \\ & =a^{\alpha_1-d} \cdot \frac{a^d\left(1-a^{50 d}\right)}{1-a^d} \end{aligned} $ $ \begin{aligned} & \Rightarrow a^{\alpha_1} \cdot \frac{\left(1-a^{50 d}\right)}{1-a^d}=3\left(2^{2 s}+1\right) …(i) \\ & f\left(a_{31}\right)=64 f\left(a_{2 s}\right) \\ & \Rightarrow a^{\alpha_1+30 d}=64 \cdot a^{\alpha_1+24 d} \\ & \Rightarrow a^{6 d}=64=2^6 \\ & \quad a^d=2 \text { ...(ii) } \end{aligned} $ Using (i) and (ii) $ \begin{aligned} & \Rightarrow \frac{a^{\sigma_1}\left(1-2^{50}\right)}{1-2}=3\left(2^{2 s}+1\right) \\ & \Rightarrow a^{\sigma_1}\left(2^{2 s}-1\right)\left(2^{2 s}+1\right)=3\left(2^{2 s}+1\right) \\ & \Rightarrow a^{\sigma_1}=\frac{3}{2^{2 s}-1} \quad \ldots \text { (iii) } \end{aligned} $ $ \begin{aligned} & \text { Now, } \sum_{i=6}^{30} f\left(a_i\right)=a^{\alpha_1-d} \sum_{i=6}^{30} a^{i d}=a^{\alpha_1-d} \cdot \frac{a^{6 d}\left(a^{2 s d}-1\right)}{\left(a^d-1\right)} \\ & =a^{\alpha_1} \cdot\left(a^d\right)^s \frac{\left|\left(a^d\right)^{2 s}-1\right|}{a^d-1} \\ & =\frac{3}{2^{2 s}-1} \times \frac{2^5 \cdot\left(2^{2 s}-1\right)}{2-1} \\ & =96.00 \end{aligned} $