Let ℝ denote the set of all real numbers. Let f: ℝ arrow ℝ be a...

Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R} be a function such that$f(x)>0$for all$x \in \mathbb{R}$, and$f(x+y)=f(x) f(y)$for all$x, y \in \mathbb{R}$. Let the real numbers$\mathrm{a}_1, \mathrm{a}_2 \ldots, \mathrm{a}_{50}$be in an arithmetic progression. If$f\left(\mathrm{a}_{31}\right)=64 f\left(\mathrm{a}_{25}\right)$, and$\sum_{i=1}^{50} f\left(a_i\right)=3\left(2^{25}+1\right)$then the value of$\sum_{i=6}^{30} f$\left(a_i\right)$ is ________