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.