Why the Sequence 8, 10, 13, 17, and 22 is Not an Arithmetic Progression

An arithmetic progression, often simply referred to as an arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant. This difference is known as the common difference. When this common difference is not constant, the sequence does not represent an arithmetic progression. In this article, we will explore why the sequence 8, 10, 13, 17, and 22 is not an arithmetic progression.

Understanding Arithmetic Progressions

Let's start by understanding the structure of an arithmetic progression. In an arithmetic sequence, the difference between consecutive terms is always the same. Mathematically, if (a_1, a_2, a_3, ldots, a_n) is a sequence, then for all (i) where (1 leq i [a_{i 1} - a_i d]

where (d) is the common difference.

Analyzing the Given Sequence

Let's examine the given sequence: 8, 10, 13, 17, and 22. We need to calculate the differences between consecutive terms, and then observe the pattern.

The first term is 8, and the second term is 10. The difference is:

[10 - 8 2]

The second term is 10, and the third term is 13. The difference is:

[13 - 10 3]

The third term is 13, and the fourth term is 17. The difference is:

[17 - 13 4]

The fourth term is 17, and the fifth term is 22. The difference is:

[22 - 17 5]

From the above calculations, we observe that the differences between consecutive terms are increasing by 1 each time: 2, 3, 4, and 5.

Conclusion

For a sequence to be an arithmetic progression, the differences between consecutive terms must be constant. However, in our given sequence, the differences are not constant. Instead, they form another sequence (2, 3, 4, 5) which is increasing by a constant amount of 1 each time. Therefore, the sequence 8, 10, 13, 17, and 22 is not an arithmetic progression.

To express it formally, if (a_n) is the (n)-th term of the sequence, we have:

[a_2 - a_1 2, quad a_3 - a_2 3, quad a_4 - a_3 4, quad a_5 - a_4 5]

Because the differences (2, 3, 4, 5) are not all the same, the sequence is not an arithmetic progression.

Additional Insights

Arithmetic progressions have several interesting properties and applications in mathematics and real-world scenarios. For example, they can be used to model linear growth or decay in various contexts, such as financial forecasting, population growth, and more.

If you are interested in exploring more about arithmetic progressions and other sequences, you can refer to textbooks or online resources that cover discrete mathematics and number theory.