Alternative Form For Fibonacci Sequence

The Fibonacci sequence 1, 1, 2, 3, 5, 8, 11... - is defined by the recurrence relation  
\[F_{n+1}=F_n+F_{n-1}\]
.Br /> In particular,
\[F_3=F_2+F_1\]

\[F_4=F_3+F_2\]

\[F_5=F_4+F_3\]

\[\vdots = \vdots+ \vdots\]

\[F_{k+1}=F_k+F_{k-1}\]

Adding these up gives
\[F_{k+1}+F_k+...+F_4+F_3=F_k+...+F_4+F_3+F_2+F_{k-1}+...+F_3+F_2+F_1\]

Many terms cancel, leaving
\[F_{k+1}=F_{k-1}+F_{k-2}F_{k-3}...+F_3+2F_2+F_1\]

\[F_2=1\]
  so we can write
\[F_{k+1}=F_{k-1}+F_{k-2}F_{k-3}...+F_3+F_2+F_1+1\]

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