Mathematics - Sequence & Series Question with Solution | TestHub

Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to \mathbb{R}$ 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 ${\mathrm{a}}_1,{\mathrm{a}}_2\dots ,{\mathrm{a}}_{50}$ be in an arithmetic progression. If $f\left({\mathrm{a}}_{31}\right)=64f\left({\mathrm{a}}_{25}\right)$, 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 ________