Let denote the set of all real numbers. Let be a function such that for all , and for all .Let the real numbers be in an arithmetic progression. If , andfind the value of\
[THOUGHT] I need to find the sum of f(a_i) from i=6 to 30 given the conditions. The function f satisfies f(x+y)=f(x)f(y) and is always positive, so it's an exponential function. The sequence a_i is arithmetic, and f(a31) = 64 * f(a25) implies f(6d) = 64 where d is the common difference, leading to b^d = 2. The total sum from i=1 to 50 is 3*(2^25 + 1), which gives a constant c = f(a1) = 3/(2^25 - 1). The sum from i=6 to 30 is then c times the sum of 2^{i-1} from i=6 to 30, which simplifies to 96. I'll verify with Python code. [/THOUGHT] [PYTHON] d25 = 2**25 denom = d25 - 1 total = 32 * (d25 - 1) # 32 is 2^5 T = 3 * total // denom print(T) [/PYTHON] [THOUGHT] The sum from i=6 to 30 of f(a_i) is 96. [/THOUGHT] \boxed{96}
[THOUGHT] The Python code computed the result as 96, which matches the derived value. The sum from i=6 to 30 is 96. [/THOUGHT] \boxed{96}