Evaluating Sigma Notations

Question

I've been having some troubling evaluating summations when the end value is not given. For example, when I am given:

(1)$$\sum_{i=0}^{n-1} {(2i+1)}$$ and

(2)$$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$

I have no idea how I would solve equation 1 without given an end value. I suppose I would have to solve for n from equation 2 first, but I also wouldn't know when to end without having an end value.

Can someone please point me towards the right direction as to how I would solve equation 1?

This is my first post ever, sorry if I am not clear with my question. Thanks!

Answer 1

HINT

Note that

$$\sum_{i=0}^{n-1} {(2i+1)}=2\sum_{i=0}^{n-1}i+\sum_{i=0}^{n-1} 1$$

and

• $\sum_{i=0}^{n-1}i=\left(\sum_{i=1}^{n} i \right)-n$
• $\sum_{i=0}^{n-1} 1=n-1$