I think I might be spoiling you because in this format it's hard to read and not likely to get accepted.

(a) c_{k}= 3c_{k-1}+ 1 for all integers (INDUCTION STATEMENT P)

c_{1}= 1

However,

c_{k-1}= 3c_{k-2}+ 1, so

c_{k}= 3 (3c_{k-2}+ 1) + 1 = (3^2)c_{k-2}+ 2 (MODIFIED INDUCTION STATEMENT P)

((Proceed inductively by replacing c_{k-m}term until you reach c_{1}= 1 ))

= 3^(k-1)c_{1}+ k

= 3^(k-1) + k

Prove this statement by general induction if needed.