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'll model the sequence (f(a_i)) as a geometric progression using the functional equation and arithmetic progression structure, then use the given conditions to find the common ratio and first term. Finally, I'll compute the required partial sum using Python for exact arithmetic. [/THOUGHT] [PYTHON] from fractions import Fraction
Given: r^6 = 64 and r>0, so r = 2
r = 2
Let f(a_i) = c * r^(i-1); sum i=1..50 equals 3(2^25 + 1)
from math import isclose
S50 = 3 * (225 + 1) c = Fraction(S50, 250 - 1) # exact rational
Sum from i=6 to 30 of f(a_i) = c * sum_{k=5}^{29} r^k = c * 2^5 * (2^25 - 1)
S_6_30 = c * (25) * (225 - 1)
print("c =", c) print("Sum i=6..30 =", S_6_30) print("As integer:", int(S_6_30)) [/PYTHON]
[THOUGHT] Using the Python-verified geometric progression parameters, the required partial sum from (i=6) to (30) is 96. [ \sum_{i=6}^{30} f(a_i) = 96 ] [/THOUGHT] \boxed{96}