I am trying to find Big O of this formula: $T(n)=T(n-1)+2n$ by using iteration however I am stuck on a step.

$T(n)=T(n-1)+2n$ I then plugged $T(n-1)$ into the equation so $T(n-1)=T(n-1-1)+2(n-1)$ which then I got $T(n)=T(n-2)+2(n-1)+2n$. $T(n-2)$ gives $T(n-3)+2(n-3)$ which plugging it back in gives $T(n)=T(n-3)+2(n-3)+2(n-1)+2n$. I substituted the following $2(n-3)+2(n-1)+2n$ into a summation that gave me $\sum_{i=0}^k 2(n-i)$$=-(k+1)(k-2n)$. I am not too sure how I should continue.


You're almost there. You correctly have $$\begin{align} T(n) &= T(n-1)+2n\\ &= [T(n-2)+2(n-1)] + 2n = T(n-2)+2(n-1)+2n\\ &= T(n-3)+2(n-2)+2(n-1)+2n\\ &= T(n-4)+2(n-3)+2(n-2)+2(n-1)+2n \end{align}$$ and in general, $$ T(n)=T(n-j)+2(n-j+1)+2(n-j+2)+2(n-j+3)+\dotsm+2n $$ Now all you need is the base value for the recurrence. Let's suppose that it was $T(0)=0$ (for the big-O estimate it really wouldn't matter if we had had, say $T(1)=3$ as our base).

To get to the base case, $T(0)$, we'd then need $n-j=0$. In that case, we'd have $$ T(n)=T(0)+2(1)+2(2)+2(3)+\dotsm+2n= \sum_{k=1}^n2k=2\cdot\frac{n(n+1)}{2} = O(n^2) $$

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    $\begingroup$ And, of course, prove that guess correct by induction. $\endgroup$ – Raphael Jan 24 '16 at 23:35
  • $\begingroup$ @Raphael. Of course. $\endgroup$ – Rick Decker Jan 25 '16 at 14:17
  • $\begingroup$ Some people seem to think it's not necessary (for a formal proof) so I have fallen into the habit of mentioning it. No offense intended. $\endgroup$ – Raphael Jan 25 '16 at 15:57

You need to find the value of $k$. What's the last possible value for $i$? If you substitute $n$ by $1$ in the original formula, what form does the term $2n$ take? You get $2*1$, which must equal $2(n-k)$. Thus, $k = n-1$.


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