User:Scythe33/temp

We know:

$$\sum_{i=1}^{n}L_i\,=\,L_{n+2}\,-3 $$

We must prove:

$$\sum_{i=1}^{n+1}L_i\,=\,L_{n+3}\,-3 $$

Recall:

$$L_{n+3}\,=\,L_{n+1}\,+\,L_{n+2}$$

Also:

$$\sum_{i=1}^{n+1}L_i\,=\,L_{n+1}+\sum_{i=1}^{n}L_i\,$$

The second equation therefore becomes:

$$L_{n+1}\,+\,\sum_{i=1}^{n}L_i\,=\,L_{n+1}\,+\,L_{n+2}\,-3 $$

and these are clearly equal, so the theorem is proven.