# Sum of the n terms of a PA

Consider the PAN finite:

(5, 7, 9, 11, 13, 15, 17, 19).

Note that:

• 5 and 19 they are extreme;
• 7 and 17 they are terms equidistant from the extremes;
• 9 and 15 they are terms equidistant from the extremes;
• 11 and 13 they are terms equidistant from the extremes.

Watch:

5 + 19 = 24 → sum of extremes

7 + 17 = 24 → sum of two equidistant terms of extremes

9 + 15 = 24 → sum of two equidistant terms of extremes

11 + 13 = 24 → sum of two equidistant terms of extremes

Based on this idea, there is the following property:

In a finite AP, the sum of two equidistant terms of extremes equals the sum of extremes.

Through this property we can find out the formula for the sum of the no terms of a PAN:

Let's consider the PAN finite . We can represent by the sum of the terms of that PAN.

Since the sum of two equidistant terms of the extremes equals the sum of the extremes, the sum of the PAN is given by the sum of the extremes times half the number of terms because in each sum two terms are involved.  So we have the formula of the sum of no terms of a PAN: • = sum of no terms
• = first term
• = nth term
• no = number of terms

Note: Through this formula we can calculate the sum of the no first terms of a PAN Whatever, just determine the number of terms we want to add.

Example 1

What is the sum of First 10 terms gives PA (1, 4, 7,…) ?

Resolution
First we have to find out what is the 10th term of this PAN:  Knowing the value of the 10th term, we can calculate the sum of the First 10 terms of that PAN: Therefore, the sum of the First 10 terms gives PA (1, 4, 7,…) é 145.

Example 2

The sum of n first positive even numbers of a PA is 132. Find the value of no.

Resolution

First, let's find out what is the nth term :  Substituting in the formula of the sum of terms: Therefore, the sum of the 11 first positive even numbers is 132.

Next: Geometric Progression