Fibonacci Squared

Number Theory Level 3

$F_{n}$ denotes the $n^{th}$ term of the Fibonacci sequence where $F_{1} = 1$ and $F_{2} = 1$ .

Given that $F_{14} = 377$ and $F_{15} = 610$

What is the value of $\displaystyle \sum_{n=1}^{15}$ $({F_{n}}^2)$

1 solution

A Former Brilliant Member
Jul 11, 2019

knowing

$F_nF_{n-1}=(F_{n-1}+F_{n-2})F_{n-1}=F^2_{n-1}+F_{n-2}F_{n-1}$

we recursively get

$F_{n}F_{n-1}=F^2_{n-1}+\dots+F_2^2+F_2F_1=F^2_{n-1}+\dots+F_2^2+F^2_1=\sum_{k=1}^{n-1}F^2_k$

Take $n=16$ to have

$F_{16}F_{15}=\sum_{k=1}^{15}F^2_k$

$F_{16}F_{15}=(F_{15}+F_{14})F_{15}=987 \times 610=602070$