Understanding and Predicting the Sequence: 14 12 10 8 6

Sequences play a significant role in mathematics and can provide valuable insights into patterns and relationships. The sequence 14, 12, 10, 8, 6 is a prime example of an arithmetic sequence, where each term is obtained by subtracting a constant value, known as the common difference, from the previous term. In this article, we will delve into the nature of this sequence, its properties, and how to predict further terms.

The Nature of Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term. The common difference can be positive, negative, or zero. In the case of the sequence 14, 12, 10, 8, 6, the common difference is -2. This means that each term is obtained by subtracting 2 from the previous term.

Identifying the Common Difference

Let's start by identifying the common difference in the given sequence:

14, 12, 10, 8, 6

The terms in the sequence are:

14 (first term) 12 (second term, obtained by subtracting 2 from 14) 10 (third term, obtained by subtracting 2 from 12) 8 (fourth term, obtained by subtracting 2 from 10) 6 (fifth term, obtained by subtracting 2 from 8)

The common difference is -2, as shown below:

14 - 12 2
12 - 10 2
10 - 8 2
8 - 6 2

Note the negative sign due to the subtraction.

Extending the Sequence

Given the common difference, we can easily extend the sequence further:

6 - 2 4 4 - 2 2 2 - 2 0 0 - 2 -2 -2 - 2 -4

Thus, the sequence can be extended as follows:

14, 12, 10, 8, 6, 4, 2, 0, -2, -4, ...

General Formula for nth Term

The nth term of an arithmetic sequence can be found using the formula:

an a1 (n - 1)d

where:

an is the nth term in the sequence a1 is the first term in the sequence d is the common difference n is the position of the term in the sequence

For our sequence:

a1 14 (first term) d -2 (common difference)

Let's apply the formula to find the 10th term:

a10 14 (10 - 1)(-2)

a10 14 9(-2)

a10 14 - 18

a10 -4

So, the 10th term of the sequence is -4.

Real-World Applications

Arithmetic sequences have numerous real-world applications. For instance, they can be used to model population growth, depreciation, and more. Consider the following scenario:

A small town has a population of 14,000 people. The population decreases by 2,000 people each year. Using the sequence 14, 12, 10, 8, 6, we can predict the population in subsequent years:

First year: 12,000 people Second year: 10,000 people Third year: 8,000 people Fourth year: 6,000 people

As we continue to apply the common difference, we can predict the population for future years.

Conclusion

The sequence 14, 12, 10, 8, 6 is a classic example of an arithmetic sequence with a common difference of -2. By understanding and applying the concept of common difference and the general formula for the nth term, we can predict future terms and apply the sequence to real-world scenarios. Whether you are a mathematics enthusiast or a data analyst, understanding arithmetic sequences can provide valuable insights and tools.