Why is 11 times the 7th term of a fibonacci series equal to the sum of 10 terms? Why is 11 times the 7th term of a fibonacci series equal to the sum of 10 terms?


I was watching scam-school on youtube the other day and this number trick just astonished me. Can someone please explain why this works?

After a lot of searching, I've been stumbling onto slightly complicated mathematical explanations. An explanation of a simpler nature, one that a child can understand, would be much appreciated.

Also, Can you extend this to find the sum of n terms of a fibonacci type sequence?
As far as I know, it seems to be nothing more than coincidence. Say you have your starting numbers, $a$ and $b$. Your ten terms are
$a,b,a+b,a+2b,2a+3b,3a+5b,5a+8b,8a+13b,13a+21b,21a+34b$

the sum of which is $55a+88b$, which just happens to $11$ times the seventh term in your sequence.
If the first term is x, and the second term is y, then
(1) x
(2) y
(3) x+ y
(4) x + 2y
(5) 2x + 3y
(6) 3x +5y
(7) 5x +8y
(8) 8x + 13y
(9) 13x + 21y
(10) 21x + 34y

The sum of all of those numbers is 55x + 88y, which factors to 11(5x + 8y)
Notice that the 7th term is 5x + 8y, so multiply the 7th term times 11 and you will get the sum of all the numbers in the sequence.