Understanding the First Three Terms of an Arithmetic Progression (AP)

When faced with the arithmetic progression (AP) formula an 2n - 1, determining the first three terms requires a systematic approach. This article aims to break down the process of finding these terms and calculating their sum.

Introduction to Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed number, called the common difference (d), to the previous term. The general form of the nth term of an AP is given by an a (n-1)d, where a is the first term and d is the common difference.

Finding the First Three Terms

Given the formula an 2n - 1, we can derive the first three terms of the AP by substituting n 1, 2, 3, respectively.

First Term (n 1)

The first term of the AP is obtained by substituting n 1: [ a_1 2(1) - 1 1 times 2 - 1 1 times 1 - 1 1 - 1 3 ]

Second Term (n 2)

The second term is found by substituting n 2: [ a_2 2(2) - 1 4 - 1 3 ]

Third Term (n 3)

The third term is determined by substituting n 3: [ a_3 2(3) - 1 6 - 1 7 ]

The resulting first three terms form the sequence: 3, 5, 7.

Sum of the First Three Terms

The sum of the first three terms can be calculated using the formula for the sum of the first n terms of an AP: Sn n/2 [2a (n-1)d]. Here, we are looking for the sum of the first three terms, so n 3.

Using the Direct Formula

The sum S3 is calculated as follows:

[begin{align*}S_3 frac{3}{2} [2a_1 (3-1) times d] frac{3}{2} [2(3) 2 times 2] frac{3}{2} [6 4] frac{3}{2} times 10 15end{align*}]

Step-by-Step Calculation

Alternatively, we can sum the terms directly:

[begin{align*}S_3 3 5 7 15end{align*}]

Conclusion

By understanding the basic structure and formula of an arithmetic progression, we can easily derive the first three terms and their sum. This method involves substituting specific values of n into the given formula to find each term and then using the sum formula or simply adding the terms. Whether you are working with the formula directly or summing the terms individually, the result for the sum of the first three terms is boxed{15}.

Keywords: Arithmetic Progression, First Three Terms, Sum of Terms