I have been trying to approach this problem for a while but I am very confused. I am really new to this topic and just trying to do some practice
Where n is an even positive integer, give a closed form for the sum:
$$\sum_{i=\frac n2}^ni=\frac n2+(\frac n2+1)+...+n$$
Hint: it's
$$(\sum_{i=1}^{n}i)-(\sum_{i=1}^{\frac n2-1}i)$$
Does that help?
Do you know (or can you work out) the closed form for the first portion above? (The closed form for the sum of the first $n$ natural numbers?)
If you don't know the most introductory example of a closed form of a summation—the sum of the first n natural numbers—I invite you to try to work it out on your own as follows:
Take a checkerboard. It has 8 rows.
In the first row put a checker on the first square. In the second row put a checker on each of the first two squares. For the $n$th row put a checker on each of the first $n$ squares. See the pattern?
Now try to guess how many checkers you have total, based on what you know of geometry and how to find the area of a triangle.
(You can do this with pencil and paper, of course; you don't need a checkerboard. The point is to work it out for yourself. You'll likely find that your first guess at a formula is slightly off. How far is it off for $n=3$? For $n=4$? So on? How much do you have to add to compensate? Why is it different from simply finding the area of a triangle?)