Thread: 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 doesn’t 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)

Originally Posted by muglyhacker

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 doesn’t 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)

HI







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Just write one more before the last one , then you should be able to see clearly.

Ah hah, thank you for that.