Calculating the Sum of an Arithmetic Progression (AP)

Arithmetic progressions (AP) are fundamental in mathematics and have numerous real-world applications. This article will guide you through the process of finding the sum of the first n terms of an arithmetic progression, given the sum of a specific number of terms and another subsequence sum.

Understanding the Sum Formula of an AP

The sum of the first n terms of an arithmetic progression is given by the formula:

Sn (n/2) × (2a (n - 1)d)

Sn a – the first term d – the common difference n – the number of terms

Given Conditions and Problem Solving

In this problem, we are given the sum of the first four terms and the sum of the first fourteen terms of an arithmetic progression:

S4 4 S14 280

Our task is to find the formula for the sum of the n terms of this arithmetic progression.

Step-by-step Solution

Step 1: Expressing the Given Sums

First, we express the given sums using the sum formula:

S4 (4/2) × (2a 3d) 2(2a 3d) 4

S14 (14/2) × (2a 13d) 7(2a 13d) 280

This simplifies to:

2a 3d 2 (Equation 1)

2a 13d 40 (Equation 2)

Step 2: Solving the System of Equations

To find the common difference d, we can eliminate the variable a by subtracting Equation 1 from Equation 2:

2a 13d - (2a 3d) 40 - 2

10d 38

d 3.8

Step 3: Finding the First Term a

Now, substituting d 3.8 into Equation 1, we get:

2a 3(3.8) 2

2a 11.4 2

2a 2 - 11.4

2a -9.4

a -4.7

Step 4: Finding the Sum of the First n Terms

With a -4.7 and d 3.8, we can express the sum of the first n terms as:

Sn (n/2) × (2(-4.7) (n - 1)(3.8))

Sn (n/2) × (-9.4 3.8n - 3.8)

Sn (n/2) × (3.8n - 13.2)

Sn (n/2) × (3.8n - 13.2)

Sn (3.8n2 - 13.2n)/2

Sn 1.9n2 - 6.6n

Thus, the sum of the first n terms of the arithmetic progression is:

Final Formula for Sn

Sn 1.9n2 - 6.6n

This formula allows you to find the sum of the first n terms of any arithmetic progression given the first term and the common difference.