Sum of Integers in Interval

You can approach this problem in a several ways:

Sum of integers from $1$ to $n$

Whenever the interval starts with $1$, it's pretty simple. You just need to use the formula below. For example, to find the sum of the integers from $1$ to $80$, inclusive...

$$\frac{n(n+1)}{2}=\frac{80 \times 81}{2}=3240$$

Sum of integers from $n$ to $m$, where $n \neq 1$

For this one, you can follow these steps. For example, what is the sum of all integers from $30$ to $101$:

1. Find the number of intergers in the interval: $101 - 30 +1 = 72$
2. Calculate the sum of the first and last number: $30+101=131$.
3. Multiply the numbers from steps 1 and 2 together and divide by $2$ (see below):

$$\frac{72 \times 131}{2}=4716$$

If you're wondering why this works, I suggest checking out the video again in the study plan.