A series of natural numbers F_1, F_2, F_3, F_4, F_5, F_6, F_7, obeys F_n+1 = F_n + F_n-1 for all integers n ≥ 2. If F_6 = 37, and F_7 = 60, then what is F_1?

GATE CSE 2023 Q3: First term of a Fibonacci-type series given F6 = 37 and F7 = 60

A series of natural numbers $F_1, F_2, F_3, F_4, F_5, F_6, F_7, \ldots$ obeys $F_{n+1} = F_n + F_{n-1}$ for all integers $n \ge 2$ .

If $F_6 = 37$ , and $F_7 = 60$ , then what is $F_1$ ?

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Answer: A (official GATE 2023 final answer key)

Concept used: Run a Fibonacci-type recurrence backwards: $F_{n-1} = F_{n+1} - F_n$ .

First step: Compute $F_5 = F_7 - F_6 = 23$ , then keep subtracting down to $F_1$ .

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Source: official GATE 2023 CS question paper and final answer key, organised by IIT Kanpur. Question paper · Answer key