[7 marks Induction and sequences. Consider the following definition of a sequence of numbers: 1 if n 0 otherwise an-1 For example, do 1, d, d2 2, ds and dOur goal is to prove the following statement: For all natural numbers n, dn> vn if and only if n is even." Complete the following two proofs, which together will prove the above statement. (a) Prove the following using induction: Yn E Z+, da-1S V2n. HINTs: in the induction step, write d2k+1 in terms of d2k-1 using the given definition of the sequence. Later as an intermediate step, use difference of squares: (2k 1(2) 4k21 (b) Prove the following (with or without using induction): Vn N, d2n V2n You may use the statement you proved in part (a) in this part