Concerning the Fibonacci sequence, use induction to prove that + F2 = F₂Fn+1 F² + F² +F² + -

Please show how the highlighted part works step by step, I dont get the substitution, how do we get ak=2k-1 and ak-1=2k-3?

A sequence (an) n>=1 is defined recursively by a1=1, a2=4, a3=9, and for n>= 4, an=an-1 - an-2 + an-3 +2(2n-3). a.) determine a4, a5, a6 b.) Conjecture a formula for an for an>= 1. c.) Prove your formula is correct using strong induction.

Use Binet's Formula to find the 42nd and 52nd term of the Fibonacci's Sequence.