Problem Analysis #18

[THOUGHT] The function (f) satisfies (f(x+y)=f(x)f(y)) and (f(x)>0), so (f(x) = e^{kx}) for some (k). The sequence (a_i = a + (i-1)d) is arithmetic. Then (f(a_i) = f(a) \cdot [f(d)]^{i-1}). Let (c = f(a)), (r = f(d)).

From (f(a_{31}) = 64 f(a_{25})), (f(a + 30d) = 64 f(a + 24d)), so (f(6d) = 64 = 2^6). Thus (r^6 = 2^6), so (r = 2) (since (r > 0)).

Hence (f(a_i) = c \cdot 2^{i-1}).

The total sum (\sum_{i=1}^{50} f(a_i) = c (2^{50} - 1) = 3(2^{25} + 1)), so (c = 3(2^{25} + 1)/(2^{50} - 1)).

Note (2^{50} - 1 = (2^{25} - 1)(2^{25} + 1)), so (c = 3 / (2^{25} - 1)).

Now (\sum_{i=6}^{30} f(a_i) = c \sum_{k=5}^{29} 2^k = c \cdot 2^5 (2^{25} - 1) = c \cdot 32 \cdot (2^{25} - 1) = [3 / (2^{25} - 1)] \cdot 32 \cdot (2^{25} - 1) = 3 \cdot 32 = 96).

To verify with Python using integer arithmetic: [/THOUGHT] [PYTHON] p = 225 numer = 3 * (p + 1) * 32 * (p - 1) denom = p2 - 1 print(numer // denom) [/PYTHON]

[THOUGHT] The Python output confirms the sum is 96. [/THOUGHT] \boxed{96}

Explanation

The reasoning follows these logical steps based on the problem's properties:

1. The functional equation (f(x + y) = f(x)f(y)) with (f(x) > 0) implies (f(x) = e^{kx}) for some constant (k), so (f(a_i)) forms a geometric sequence for the arithmetic progression (a_i = a + (i-1)d).

2. Specifically, (f(a_i) = c \cdot r^{i-1}), where (c = f(a)) and (r = f(d) > 0).

3. From (f(a_{31}) = 64 f(a_{25})), substitute indices: (f(a + 30d) = 64 f(a + 24d)), so (f(6d) = 64 = 2^6). Thus, (r^6 = 2^6), and since (r > 0), (r = 2).

4. Therefore, (f(a_i) = c \cdot 2^{i-1}).

5. The given sum (\sum_{i=1}^{50} f(a_i) = c \sum_{i=0}^{49} 2^i = c (2^{50} - 1) = 3(2^{25} + 1)).

6. Factorize: (2^{50} - 1 = (2^{25} - 1)(2^{25} + 1)), so (c = \frac{3(2^{25} + 1)}{(2^{25} - 1)(2^{25} + 1)} = \frac{3}{2^{25} - 1}).

7. Now compute (\sum_{i=6}^{30} f(a_i) = c \sum_{i=6}^{30} 2^{i-1} = c \sum_{k=5}^{29} 2^k = c \cdot 2^5 (2^{25} - 1) = c \cdot 32 (2^{25} - 1)).

8. Substitute (c): (\frac{3}{2^{25} - 1} \cdot 32 (2^{25} - 1) = 3 \cdot 32 = 96).

9. Python verification used integer arithmetic: p = 2**25; numer = 3 * (p + 1) * 32 * (p - 1); denom = p**2 - 1; print(numer // denom), yielding 96, confirming the result.