Fibonacci Day: Adding the Evens

If you look closely, you'll find that today is in fact a Fibonacci day. 13 is the seventh Fibonacci number, so we're in for a treat!

Last month, we added up the Fibonacci numbers. This time, let's add the Fibonacci numbers in the even positions. We'll list some out.

1 = 1
1 + 3 = 4
1 + 3 + 8  = 12
1 + 3 + 8 + 21 = 33

See the pattern? Same thing as last time! Just the odd spots minus one!

1 = 1 = 2 - 1
1 + 3 = 4 = 5 - 1
1 + 3 + 8  = 12 = 13 - 1
1 + 3 + 8 + 21 = 33 = 34 - 1

Why does this work? Let's split each number into the previous two Fibonacci numbers.

1 + 3 + 8 + 21 = 33 = 34 - 1

1 + (1 + 2) + (3 + 5) + (8 + 13) = 34 - 1

Because of the associative property, we can eliminate the parentheses, giving us what we did last time, adding all the Fibonacci numbers. That would always give us a Fibonacci number minus one as well.

If you know something cool about Fibonacci numbers, please let me know!