<< Was anybody surprised by Fibonacci[6*n]*Fibonacci[n]/Fibonacci[3*n]/Fibonacci[2*n] - Fibonacci[3*n]/Fibonacci[n] ? NeilB found a short Mma proof. >> Fred> I suppose it is a little unexpected that the answer is not exactly zero ... We still have the nice little puzzle of how NeilB got Mma to say -2*(-1)^n. He then quickly found In[149]:= FunctionExpand[ Fibonacci[10*n]*Fibonacci[n]/Fibonacci[2*n]/Fibonacci[5*n] - Fibonacci[8*n]/Fibonacci[4*n] + (-1)^n* Fibonacci[4*n]/Fibonacci[2*n], n \[Element] Integers] Out[149]= 1 FullSimplify instead of FunctionExpand also works, but slower. Oddly, discharging both barrels at once, In[146]:= FullSimplify[ FunctionExpand[ Fibonacci[10*n]*Fibonacci[n]/Fibonacci[2*n]/Fibonacci[5*n] - Fibonacci[8*n]/Fibonacci[4*n] + (-1)^n* Fibonacci[4*n]/Fibonacci[2*n]], n \[Element] Integers] Out[146]= ((-(-1)^n)*4^(5*n) + (1 + Sqrt[5])^(10*n))/ (-(-4)^(5*n) + (1 + Sqrt[5])^(10*n)) A point from which Mma seems unable to reach 1. ? --rwg