1. mathematical problem in Physics

The total work done by the force as the object moves from position x1 to xn can be approximated by the formula

What is meant by (i) in this?
Is it first middle or last value? Below the sigma it is (i=1) but in graph it is in middle.
kindly explain.

http://www.staff.amu.edu.pl/~romangoc/M7-work-variable-force.html

2. The i value runs from 1 and n (in this example) and you substitute whatever i value you're on into the part to the right of the $\sum$ symbol.

The $\sum$ symbol is a way of simplifying a long addition problem.

Lets do an example using the formulas you've been given.
Lets say n=5. This means that there will be 5 values of $\Delta W$ to consider.

Let the first one be called $\Delta W_1$, the second $\Delta W_2$ etc up to $\Delta W_n$.

Then what $\sum_{i=1}^n \Delta W_i$ does is we start at i=1 and add together all $\Delta W$ values from $\Delta W_1$ up to $\Delta W_n$ which in this case is 5.

So we get $\sum_{i=1}^n \Delta W_i = \sum_{i=1}^5 \Delta W_i = \Delta W_1 + \Delta W_2 + \Delta W_3 + \Delta W_4 + \Delta W_5$.