Understanding the Common Difference in an Arithmetic Sequence: A Detailed Guide

Arithmetic sequences are fundamental in mathematics, forming the backbone of various mathematical and computational problems. One key concept in arithmetic sequences is thecommon difference. This article aims to explain the common difference in the context of an arithmetic sequence where the nth term is given by the formula 16 - 6n. We will delve into the methodology and explore step-by-step how to find the common difference and relate it to the nth term.

What is an Arithmetic Sequence?

Before diving into the common difference, it's essential to understand what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is known as the common difference.

The Given nth Term

The nth term of the given arithmetic sequence is specified by the formula:

16 - 6n

Understanding the Sequence

Let's start by generating the first few terms of the sequence:

Calculating the First Few Terms

First term (n 1):
16 - 6(1) 10

Second term (n 2):
16 - 6(2) 4

Third term (n 3):
16 - 6(3) -2

Fourth term (n 4):
16 - 6(4) -8

Thus, the sequence looks like this:

10, 4, -2, -8, ...

Calculating the Common Difference

The common difference is the difference between any two consecutive terms in the sequence. Let's calculate it for the given sequence:

Step-by-Step Calculation

Calculate the second term (n 2): 16 - 6(2) 4. Calculate the first term (n 1): 16 - 6(1) 10. Subtract the first term from the second term to find the common difference: 4 - 10 -6.

Alternatively, we can express the common difference by calculating it between any two consecutive terms using the given formula:

Using the nth Term Formula

The nth term is 16 - 6n, and the (n - 1)th term is 16 - 6(n - 1) 16 - 6n 6 22 - 6n.

To find the common difference, subtract the (n - 1)th term from the nth term:

(16 - 6n) - (22 - 6n)  -6

This confirms that the common difference is -6.

Example and Verification

To further verify, let's check the common difference between the third and second terms:

-2 - 4  -6

This again confirms that the common difference is -6.

Conclusion and Applications

Arithmetic sequences are widely used in various fields, including finance, physics, and computer science. Understanding the common difference helps in solving problems related to patterns, predictions, and modeling. The given problem illustrates the process of finding the common difference for an arithmetic sequence defined by the formula 16 - 6n.

For further learning, you can explore the concept of the sum of terms in an arithmetic sequence, as well as related topics like geometric sequences and their differences.