Understanding the Fifth Term of the Sequence 2, 5, 10, 17, 26: A Comprehensive Guide

Mathematics can indeed be a pleasure, as many of us agree. Let's dive into the fascinating world of number sequences and explore how to find the fifth term of the sequence 2, 5, 10, 17, 26. We will also discuss the underlying mathematical principles and provide a clear explanation of the sequence's pattern.

Understanding the Sequence

The given sequence (2, 5, 10, 17, 26) can be recognized as a quadratic sequence, where each term is generated by adding consecutive odd numbers to the previous term. Specifically, the sequence can be described as:

x[n] x[n-1] 2n - 1

Let's break this down step by step:

Step-by-Step Analysis

First Term (n1): x[1] 2 Second Term (n2): x[2] x[1] 2(2) - 1 2 3 5 Third Term (n3): x[3] x[2] 2(3) - 1 5 5 10 Fourth Term (n4): x[4] x[3] 2(4) - 1 10 7 17 Fifth Term (n5): x[5] x[4] 2(5) - 1 17 9 26

Thus, the fifth term of the sequence is 26.

Quadratic Sequence and Second Differences

A quadratic sequence is defined by the second differences remaining constant. Let's examine the differences in the sequence:

Differences between terms: 5 - 2 3, 10 - 5 5, 17 - 10 7, 26 - 17 9 Second differences: 5 - 3 2, 7 - 5 2, 9 - 7 2

The second differences (2) are constant, confirming that the sequence is quadratic.

Mathematical Formulas

The sequence can also be represented by the formula x[n] n^2 - 1. Let's demonstrate:

x[1] 1^2 - 1 1 - 1 2 x[2] 2^2 - 1 4 - 1 5 x[3] 3^2 - 1 9 - 1 10 x[4] 4^2 - 1 16 - 1 17 x[5] 5^2 - 1 25 - 1 26

This confirms that the fifth term of the sequence is indeed 26.

Conclusion

Understanding the fifth term of the sequence 2, 5, 10, 17, 26 involves recognizing the quadratic nature of the sequence and applying the appropriate mathematical formula. The sequence follows the pattern of adding consecutive odd numbers, and the second differences being constant validates the quadratic relationship.

By exploring this sequence, we not only gain a deeper understanding of mathematical sequences but also appreciate the elegance and patterns in numbers. Mathematics truly is a pleasure to explore.