Easy: Solve T(n)=T(n-1)+n by Iteration Method

Question

Can someone please help me with this ?

Use iteration method to solve it. T(n) = T(n-1) +n

Explanation of steps

Answer 1

Easy Method:

T (n) = T (n - 1) + (n )-----------(1)
 //now submit T(n-1)=t(n)

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

now submit (2) in (1) you will get
i.e T(n)=[T(n-2)+n-1]+(n)
T(n)=T(n-2)+2n-1 //simplified--------------(3)

 now, T(n-2)=t(n)
T(n-2)=T((n-2)-2)+[2(n-2)-1]
  T(n-2)=T(n-4)+2n-5---------------(4)
  now submit (4) in (2) you will get
   i.e T(n)=[T(n-4)+2n-5]+(2n-1)
  T(n)=T(n-4)+4n-6 //simplified
    ............
 T(n)=T(n-k)+kn-6
  **Based on General form T(n)=T(n-k)+k, **
  now, assume n-k=1 we know T(1)=1
            k=n-1

    T(n)=T(n-(n-1))+(n-1)n-6
    T(n)=T(1)+n^2-n-10
   According to the complexity 6 is constant

         So , Finally O(n^2)