recreational mathematics - 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?

Answer

@Claude Leibovici

In fact, there is a different way to answer this question using characteristic polynomials.

All Fibonacci-like sequences are associated with the same characteristic polynomial $x^2-x-1$ due to their common property : $$\psi_{n+2}-\psi_{n+1}-\psi_{n}=0.$$

Let us define a new sequence in the following way :
$$\chi_n:=(\psi_{n+1}+\psi_{n+2}+...+\psi_{n+10})-11 \psi_{n+7}. \tag{1}$$

We want to show that, for any $n \geq 0$, $\chi_n=0$.

This is an easy consequence of the fact that the characteristic polynomial of sequence $\chi_n$, i.e.,

$$p(x):=(x+x^2+...+x^9+x^{10})-11x^7$$

is divisible by $(x^2-x-1).$

Precisely :

$$p(x)=x(x^2 - x - 1)(x^7 + 2x^6 + 4x^5 - 4x^4 + x^3 - 2x^2 - 1).$$