Finding the Sum of the First 7 Terms of an Arithmetic Progression Given the 4th Term

Arithmetic progressions (AP) are a fundamental concept in mathematics, often appearing in various fields such as finance, physics, and computer science. One common problem involves finding the sum of the first few terms of an AP given certain conditions. In this article, we will walk through a step-by-step solution to finding the sum of the first 7 terms of an AP, given that the 4th term is 8. We will also explore alternative methods and provide a thorough explanation of the process.

Arithmetic Progression Basics

An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference (d), to the previous term. The general formula for the nth term of an arithmetic progression is given by:

a_n a (n-1)d

where a is the first term, d is the common difference, and n is the term number. In this particular problem, we are given that the 4th term is 8.

Given Information and Step-by-Step Solution

Let's denote the first term of the AP by a and the common difference by d. According to the given information, the 4th term is 8. Thus, we can write:

a 3d 8

We want to find the sum of the first 7 terms of the AP. The formula for the sum of the first n terms of an AP is:

S_n n/2 [2a (n-1)d]

For n 7, the sum of the first 7 terms is:

S_7 7/2 [2a 6d]

Let's express 2a 6d in terms of the given equation. From the equation a 3d 8, we can express a as:

a 8 - 3d

Substituting a 8 - 3d into the sum formula:

2a 6d 2(8 - 3d) 6d 16 - 6d 6d 16

Now, substituting this result into the sum formula:

S_7 7/2 × 16 56

Thus, the sum of the first 7 terms of the arithmetic progression is:

strongspan style"font-size: 18px;"boxed{56}/span/strong

Alternative Methods

Another approach to solving this problem is to list out the terms of the AP directly and add them up. If the 4th term is 8, then the terms of the AP can be represented as:

T1 a 8 - 3d

T2 8 - 2d

T3 8 - d

T4 8

T5 8 d

T6 8 2d

T7 8 3d

Summing these terms, we get:

(8 - 3d) (8 - 2d) (8 - d) 8 (8 d) (8 2d) (8 3d) 56

As each pair of terms (8 - d and 8 d, 8 - 2d and 8 2d, etc.) cancels each other out, the sum remains:

56

Conclusion

In this article, we have explored the process of finding the sum of the first 7 terms of an arithmetic progression given that the 4th term is 8. We have provided a detailed step-by-step solution and an alternative method to solve the problem. The key takeaway is that understanding the basic formulae and applying them systematically can lead to accurate solutions. Whether you use the algebraic method or the list approach, the sum of the first 7 terms of the arithmetic progression is 56.