# Questions concerning $\Omega$ and $\Theta$

How do I solve this: $\displaystyle\sum_{k = 1}^n \frac{k}{3}$ is $\Omega(n^2)$? I know that the summation would be $\displaystyle\sum\limits_{k=1}^n \dfrac{n^2+n}{6}$, but how do I solve $\displaystyle\sum\limits_{k = 1}^n \frac{k}{3}$ is $\Omega(n^2)$?

Also how do I solve $\sum\limits_{j = 1}^n\sum\limits_{k=1}^n 7n$ is $\Theta(n^3)$? Does anyone know any good website explaining these type of problems?

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Your sums seem to be miswritten. Can you clarify what you mean by $\sum_{i=1}^n k/3$? – Ian Coley Mar 26 '13 at 4:06
sorry i will fix that – user1688953 Mar 26 '13 at 4:11
Your fourth sum could use some work, unless there's just supposed to be a $1$ there. – Julien Clancy Mar 26 '13 at 4:36
thanks, just fixed it – user1688953 Mar 26 '13 at 4:50
Are you sure it's correct now? As it's written, the sum is in fact not Theta of $n^2$. – Julien Clancy Mar 26 '13 at 5:56

You're right in that $\sum_1^n \frac{k}{3} = \frac{n^2 + n}{6}$. Recall that $f \in \Omega(g)$ if there are some $c > 0$ and $x_0$ with $f(x) > cg(x)$ for all $x > x_0$. Intuitively, $f$ is $\Omega(g)$ if $g$ bounds $f$ below. In this case you can just take $c = \frac{1}{6}$, assuming $n > 0$. The strategy here is simply to find the right constant.
Recall that $f \in \Theta(g)$ if there are $c, C > 0$ and $x_0$ with $cg(x) < f(x) < Cg(x)$ for all $x > x_0$, or equivalently if it is both $O(g)$ and $\Omega(g)$. You can pick up the computability part from here.