Question

Solve the following recurrences by repeatedly unrolling them, aka the method of substitution. You must show your work, otherwise you will lose points. Assume a base case of T(1) = 1. As part of your solution, you will need to establish a pat- tern for what the recurrence looks like after the k-th iteration. You must to formally prove that your patterns are correct via induction. Your solutions may include integers, n raised to a power, and/or logarithms of n. For example, a solution of the form 8log2 n is unacceptable; this should be simplified as n1og2 8 = n3. i. T(n) 8T(n/2)n3 ii. T(п) — 67 (п/3) + п

Answer 1

D T(): 87(43) + m3 -0 (3)- 87(7.) + ma co putting value of T(A) in equation- T(n)= 8/87 (24) + 2 } ] + 23 T(" 8*7 (3) + né+os Toe he fet (3) ****** 7(2* pos 1 (43) + ntodens The= (nta? tadt ---) long as times To)= negr = 10 (n legn) 0.7”)= 6T (3) + T(n)= 6f 67(33) +}]+ T(") = 6* T () + 20+) T[")= 6? [671) + * * * T(m): 67(3) +49 + ento Timanfit24+6 --- ] ly sº times TC)= " ( 24759-17 -> (1732-,) s bilang sets)_D => on lry 3**-n +2n+n 2-)