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Let F_1 = 1, F_2 = 1 and F_{n+2} = F_{n+1} + F_n be the Fibonacci…
Question
Let F_1 = 1, F_2 = 1 and F_{n+2} = F_{n+1} + F_n be the Fibonacci numbers. The infinite sum sum_{n=1}^{infinity} F_n/3^n equals m/n in lowest terms. Find m + n.
✓ Verified answer: 8checked by our engine — not a guess
Step-by-step solution
Let G(x) = sum_{n>=1} F_n x^n. From the recurrence, G(x) = x/(1 - x - x^2).
Set x = 1/3 (within radius of convergence since 1/3 < 1/phi):
G(1/3) = (1/3)/(1 - 1/3 - 1/9) = (1/3)/((9 - 3 - 1)/9) = (1/3)/(5/9) = (1/3)*(9/5) = 3/5.
m/n = 3/5, so m + n = 8.
Final answer8
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