Problem of the Week 1097

Five by Five

The Fibonacci recurrence (F₁ = 1, F₂ = 1 and F_n = F_{n - 1} + F_{n - 2}) leads to an infinite sequence of numbers starting with

1, 1, 2, 3, 5,
8, 13, 21, 34, 55,
89, 144, 233, 377, 610, ...

Note that the 5th, 10th and 15th numbers are all divisible by 5.

Show that every fifth number in the sequence is divisible by 5.

© Copyright 2008 Andrew Beveridge and Stan Wagon. Reproduced with permission.