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The problem is as stated:

Prove that \(\displaystyle F_1*F_2+F_2*F_3+...+F_{2n-1}*F_{2n}=F^2_{2n}\)

But earlier in my text I proved by induction that \(\displaystyle F_{2n}=F_1+F_2+...+F_{2n-1}\). Do I need to use this earlier proof in my current proof. I tried adding \(\displaystyle F_{2n+1}F_{2n+2}\) to the right and left hand side of the first equation and tried to find \(\displaystyle F_{2n+1}F_{2n+2}+F^2_{2n}=F^2_{2n+2}\) but that doesn't seem to be going anywhere. (Why doesn't that seem to work in this case? Because I am multiplying two sums together?)

Am I wrong in assuming that I am supposed to prove this by induction?

Prove that \(\displaystyle F_1*F_2+F_2*F_3+...+F_{2n-1}*F_{2n}=F^2_{2n}\)

But earlier in my text I proved by induction that \(\displaystyle F_{2n}=F_1+F_2+...+F_{2n-1}\). Do I need to use this earlier proof in my current proof. I tried adding \(\displaystyle F_{2n+1}F_{2n+2}\) to the right and left hand side of the first equation and tried to find \(\displaystyle F_{2n+1}F_{2n+2}+F^2_{2n}=F^2_{2n+2}\) but that doesn't seem to be going anywhere. (Why doesn't that seem to work in this case? Because I am multiplying two sums together?)

Am I wrong in assuming that I am supposed to prove this by induction?

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