Compute n=1∑∞FnFn+21,where Fn denotes the nth Fibonacci number, so F0=0 and F1=1.
Solution — click to reveal
We can get the sum to telescope by writing FnFn+21=FnFn+1Fn+2Fn+1.Since Fn+1=Fn+2−Fn,FnFn+1Fn+2Fn+1=FnFn+1Fn+2Fn+2−Fn=FnFn+11−Fn+1Fn+21.Then n=1∑∞FnFn+21=(F1F21−F2F31)+(F2F31−F3F41)+(F3F41−F4F51)+⋯=F1F21=1.