# Is this possible?

Is it possible to convert this forumla $$r(\theta) = \sum_{n=0}^\theta\left(\frac{2n+1}{2}\right)$$ to one without the $\sum$ sign? If so, how?

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Make use of the following summations: $$\sum_{n=0}^{\theta} 1 = \theta+1$$ $$\sum_{n=0}^{\theta} n = \dfrac{\theta(\theta+1)}2$$
Hence, we have $$r(\theta) = \sum_{n=0}^{\theta}n + \dfrac12 \sum_{n=0}^{\theta} 1 = \dfrac{\theta(\theta+1)}2 + \dfrac{\theta+1}2 = \dfrac{(\theta+1)^2}2$$
Yes. You want to use the formula for the sum of an Arithmetic Series. Note that the original sum is $$\displaystyle\sum_{n=0}^{\theta} \dfrac {2n+1}{2} = \dfrac {1}{2} \cdot \displaystyle\sum_{n=0}^{\theta} (2n+1).$$It is easy to prove, using the formula I linked you to, that $$\displaystyle\sum_{n=0}^{\theta} (2n+1) = (n+1)^2.$$ This is a well-known fact. Now finish up.