Ratio of all Fibonacci terms to every third one

Question

With $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ , evaluate the ratio $\frac{\sum_{i=1}^{300}F_i}{\sum_{j=1}^{100}F_{3j}}.$

Short answer

$2$ . Group the numerator in triples: $F_{3k-2}+F_{3k-1}+F_{3k}=F_{3k}+F_{3k}=2F_{3k}$ . So the numerator is exactly twice the denominator.

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Lessons behind this question

Counting and Combinatorics
Applications: Difference Equations, Markov Chains, and Fibonacci

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