Fibonacci series cancellation

I am currently re-viewing some text books of a course I took a few years ago and have come across this Fibonacci relation in the form *Fn = Fn+2 Fn+1*

*F0 = F2 - F1*

*F1 = F3 - F2*

*F2 = F4 - F3*

*
*

*Fn = Fn+2 Fn+1*

Adding LHS together we get LHS = *F0 + F1 + F2 +
+ Fn *

and RHS =* F2 - F1 + F3 - F2 + F4 - F3 +
+ Fn+2 Fn+1*

And using telescoping cancellation the book states that the RHS = *Fn+2 **- 1*

I can only get the RHS to *=* *Fn+2 Fn+1 - 1 (Because F1 = 1). *I am stuck with the * Fn+1 *term of the RHS; nothing I can see cancels it out, yet clearly the book states that it has gone. Unfortunately the book doesnt clearly show the cancellation in steps. Anyone here like to help, am I being very dumb???

(sorry about the formatting; I subscripted all the numbers but it doesn't seem to cut and paste into this)