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] Because (a_i) is an arithmetic progression and (f(x+y)=f(x)f(y)), the values (f(a_i)) form a geometric progression: (f(a_i)=t r^{i-1}) where (t=f(a_1)) and (r=f(d)). Use (f(a_{31})=64 f(a_{25})) to find (r), use the total sum to find (t), then compute (\sum_{i=6}^{30} f(a_i)). I will verify all algebra with Python. [/THOUGHT] [PYTHON] from fractions import Fraction
r = 2 # from r^6=64 t = Fraction(3, 225 - 1) S_6_30 = t * sum(2(i-1) for i in range(6, 31)) t, S_6_30 print(int(S_6_30)) [/PYTHON]
[THOUGHT] The computed value of (\sum_{i=6}^{30} f(a_i)) is 96. [ \boxed{96} ]