The following recurrence relations follow the form of the Master Method. Solve each.
6.126 Prove that the recurrences T(n) = aT(n/b) + c * n^k and T(1) = d and S(n) = aS(n/b) + n^k and S(1) = 1 have the same asymptotic solution, for any constants a ≥ 1, b > 1, c > 0, d > 0, and k ≥ 0.
6.111 T(n) = 2^(n/2) + n
6.118 T(n) = 2T(n/2) + n
6.119 T(n) = 2T(n/4) + n^2
6.120 T(n) = 2T(n/4) + n
6.121 T(n) = 4T(n/2) + n^2
6.122 T(n) = 4T(n/2) + n
6.123 T(n) = 4T(n/4) + n^2
6.124 T(n) = 4T(n/4) + n
6.126 Prove that the recurrences T(n) = T(n) + cn and T(1) = d and S(n) = S(n) + n and S(1) = 1 have the same asymptotic solution, for any constants a ≥ 1, b > 1, c > 0, d > 0, and k ≥ 0.
6.127 Consider the Master Method recurrence T(n) = aT(n) and T(1) = 1. Using induction, prove the summation from the proof of the Master Theorem: prove that