渐近上界

Question

Hi I solved a question with recursion tree method. Then I reached the below equatition. n ∑ 3^(i-1)(n - (i - 1)) i=1

I need to find asymptotic upper bound for that equation. Any help would be appreciated.

Answer 1

Wolfram Alpha is a great tool for this: https://www.wolframalpha.com/input/?i=sum(3%5E(i-1)(n+-+i+%2B+1)+for+i+%3D+1..n)

That tool simplifies the sum to: (-2n + 3^(n+1) - 3)/4 .

In terms of big-O, that's O(3^n).

Answer 2

Let u(n) = 3^(n-1) + 2*3^(n-2) + ... + n, then
u(n+1) = (3^n + 3^(n-1) + ... + 1) + 3^(n-1) + 2*3^(n-2) + ... + n = (3^(n+1)-1)/2 + u(n) = 3*u(n) + n + 1 u(n) = (3^(n+1) - 2n - 3) / 4.