Let T(n) be defined by T(1) = 10 and T(n + 1) = 2n + T(n) and for all integers n ≥ 1 . Which of the following represents the order of growth of T(n) as a function of
(A)
O(n)
(B)
O(n log n)
(C)
O(n2)
(D)
O(n3)
Answer: (C)
Explanation:
T(n + 1) = 2n + T(n) By substitution method: T(n + 1) = 2n + (2(n-1) + T(n-1)) T(n + 1) = 2n + (2(n-1) + (2(n-2) + T(n-2))) T(n + 1) = 2n + (2(n-1) + (2(n-2) + (2(n-3) + T(n-3)))) T(n + 1) = 2n + 2(n-1) + 2(n-2) + 2(n-3)......2(n-(n-1) + T(1)) T(n + 1) = 2n + 2n - 2 + 2n - 4 + 2n - 6 +.... + 10 T(n + 1) = 2[n + n + n + ...] - 2[1 + 2 + 3 +...] T(n + 1) = 2[n*n] - 2[n(n+1)/2] T(n + 1) = 2[n*n] - [n*n + n] T(n + 1) = n*n - n T(n + 1) = O(n2)
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