Trapezoidal Rule/ Having trouble understanding what is k?! QUESTION:
I know of the more understanding formula for the trapezoidal rule. However, I came across this new form in a book i'm reading. Can someone tell me how I'm suppose to enter the respected values into this new form. Really would like to understand what is going on.

I'm assuming that the first part (b-a)/n will come to 1 so we can disregard it. 
Next I'm assuming that [f(a) + f(b)]/2 will be set up like (f(-2)+f(5))/2 which is about 3.667544
The summation part is confusing and I'm not even for sure if i'm correct above. I'm coming out with an answer around 13.334698. Which it should be around 17.017422. Any guidance is appreciated. 

It is the ordinary Trapezoidal Rule. You may have seen the version 
$$\frac{\Delta x}{2}\left(f(x_0)+2f(x_1)+2f(x_2)+\cdots+2f(x_{n-1})+f(x_n)\right),$$
where $x_k=a+k\Delta x$ and $\Delta x=\frac{b-a}{n}$.
In the formula you quoted, the first and last terms, which are $f(a)$ and $f(b)$ respectively, have been separated out and put in front, as $\frac{f(a)+f(b)}{2}$. 
The rest of the terms, corresponding to $k=1$ to $n-1$, when divided by $2$, give $f(x_1)+f(x_2)+\cdots+f(x_{n-1})$, exactly as in the quoted formula. 
