Substitution method for solving recurrences

Question

First of all sorry for asking such a basic question.

But I am having difficulties understanding substitution method for solving recurrences.I am following Introduction to Algo.s -CLRS. As I am not able to find enough examples and ambiguity is the main concern.Especially the induction step.In the text books we have to prove f(n) implies f(n+1) but in CLRS this step is missing or may be I am not getting the example. Please explain step by step how to prove that O(n^2) is the solution for recurrence function T(n)=T(n-1)+n

Its the general steps of substitution method that I want to understand. If you could shed some light on strong mathematical induction and provide links to material on substitution method that'll be helpful also.

Answer 1

In substitution method, simply replace any occurance of T(k) by T(k-1) + k, and do it iteratively.

T(n) = T(n-1) + n = 
= (T(n-2) + (n-1)) + n = 
= T(n-3) + (n-2) + (n-1) + n = 
= ...  =
= 1 + 2 + ... + n-1 +  n

From sum of arithmetic progression, you can get that T(n) is in O(n^2).

Note that substitution method is usually used to get an intuition on what the complexity is, to formally prove it - you will probably need a different tool - such as mathematical induction.

The formal proof will go something like that:

Claim: T(n) <= n^2
Base: T(1) = 1 <= 1^2
Hypothesis: the claim is true for each `k < n` for some `n`.
T(n) = T(n-1) + n <= (hypothesis) (n-1)^2 + n = n^2-2n + 1 < n^2 (for n > 1)