I am using Mathematica to explore the properties of the Fibonacci Sequence. Below is the functions I have defined
H[n_] := \[Phi]^n/Sqrt[5]
F[n_] := Round[H[n]]
S[m_] := Sum[F[n], {n, 1, m}]
Through experimentation I have realised that S[m]=SF[m-1]+F[m]
. How do I obtain a conjectured formula for this and prove it by induction?
Please let me know if you need me to clarify anything further.
Fibonacci[]
is built-in? Try this:FullSimplify[Sum[Fibonacci[k], {k, 1, n}] == Fibonacci[n + 1] + Fibonacci[n] - 1, Element[n, Integers]]
$\endgroup$SF[m]
? $\endgroup$