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.