How to evaluate the sum $\sum _{k=i+j-1}^n\:1$

Question

How would I go about solving this summation:

$$\sum_{k=i+j-1}^n\:1$$

Answer 1

You're summing over $n-(i+j-1)+1=n-i-j+2$ terms, where all the terms are $1$. so the required sum is simply $n-i-j+2$.