Sum of the Fibonacci Numbers 1 1 2 3 5 8 Fn Fn 1

Sum of the Fibonacci Numbers 1 + 1 + 2 + 3 + 5 + 8 + ... + F_n = F_n + 1 - 1 What is the smallest counter example: What is the next smallest number: How is the next smallest number related to the smallest counter example(there will be some algebra here):

Solution

F0 =0

0 = 1-1 and hence F0 = F2-1

F1 = 1 = F3-1

F2 = 1 not equal to F4-1

Hence 2 is the smallest positive integer which contradicts the given statement.

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Next smallest number = 3 since F3 not equals F5-1

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Smallest number +1 = Next smallest number is the relation