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 a_1, a_2, . . ., a_{50} be in an arithmetic progression. If f ( {\a}_{31}) =64 f ( {\a}_{25}) and \sum_{i=1}^{50} f (a_i) =3 (2^{25}+1) then the value of \sum_{i=6}^{30} f (a_i) is _____ [JEE Adv 2025 P1]

Functions

Section: Mathematics

Question : 4 of 30

Marks: +1, -0

Let ℝ denote the set of all real numbers. Let f:ℝ⟶ℝ be a function such that f(x)>0 for all x∈ℝ, and f(x+y)=f(x)f(y) for all x,y∈ℝ. Let the real numbers a1,a2,...,a50 be in an arithmetic progression. If f(a31)=64f(a25) and

∑

i=1

f(ai)=3(225+1) then the value of

∑

i=6

f(ai) is _____

[JEE Adv 2025 P1]

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