Question

Big-O notation.

Let

T(n) be given using the recursive formula.

T(n) = T(n-1) + n,

T(1) = 1.

Prove that T(n) = O(n²).

Answer 1

T(n) = T(n-1) + n,
T(1) = 1.

using substitution method:
---------------------------
T(n)
= T(n-1) + n
= T(n-2) + n-1 + n
= T(n-3) + n-2 + n-1 + n
= T(1) + 2 + ... + n-2 + n-1 + n
= 1 + 2 + ... + n-2 + n-1 + n
This is sum of first n natural numbers. It's formula is n(n+1)/2.
= n(n+1)/2
= (n2+n)/2
we can ignore lower order terms(n) and constant factors(/2).
so, time complexity is O(n2)
T(n) = O(n2)