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I am practicing review problems to practice from what what we last learnt in lecture, and I admit I am very lost.

I have no idea how to start these sort of problems

Prove for any $n \geq 1: F_1 + F_3 + F_5 + \cdots + F_{2n-1} = F_{2n}$ where $F_n$ is the $n$th term Fibonacci number.

I would greatly appreciate this answered so I can use it as a base to continue and answer the other problems myself. Or, if you would prefer instead of giving a full answer, please feel to give hints/advice to point me in the right direction.

Thank you.

Patrick Da Silva
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user3248
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1 Answers1

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Since $F_n$ is a sequence defined recursively, you have $F_{2n} = F_{2n-1} + F_{2n-2}$. It is therefore a great idea to think about induction. Try to inspire yourself from $$ F_1 + F_3 + \cdots + F_{2n-3} + F_{2n-1} = F_{2n} = F_{2n-1} + F_{2n-2}. $$

Hope that helps,

Patrick Da Silva
  • 41,413