= Fibonacci addition formula
{c}
{title2=$F_{n+k}=F_kF_{n+1}+F_{k-1}F_n$}
With the standard <Fibonacci numbers> $F_0=0$, $F_1=1$, this identity holds for $n\ge0$, $k\ge1$. At $k=1$ it is immediate. If it holds for $k$ at every $n$, apply it at $n+1$ and use $F_{n+2}=F_{n+1}+F_n$ to obtain $F_{n+k+1}=F_{k+1}F_{n+1}+F_kF_n$, proving the assertion by <mathematical induction>. Taking $k=n$ gives $F_{2n}=F_n(F_{n+1}+F_{n-1})$ for $n\ge1$, linking the doubling formula to the <Lucas numbers>.
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