Decoding the Sequence: Finding the Next Term in a Quadratic Sequence

Diving into the fascinating world of number sequences, today we explore a specific sequence and uncover the pattern that governs its progression. This particular sequence is: 2, 5, 10, 17, 26, ___, and we aim to find the number that comes next.

Understanding the Pattern: A Step-by-Step Approach

Let's break down the sequence and uncover the underlying pattern step-by-step. The given sequence is:

2 5 10 17 26 x

The simplest way to understand the pattern is by observing the differences between consecutive terms. Here's what you'll find:

5 - 2 3 10 - 5 5 17 - 10 7 26 - 17 9

We can observe that the difference between consecutive terms increases by 2 each time. This pattern suggests a quadratic relationship among the numbers. To confirm, let's express the sequence in a general form.

Identifying the Quadratic Formula

The general term of a quadratic sequence can be expressed as:

tn an2 bn c

Given the first few terms, we can derive the coefficients:

For n 1, t1 2 For n 2, t2 5 For n 3, t3 10

Using these terms, we can set up the following equations:

2 a(12) b(1) c 5 a(22) b(2) c 10 a(32) b(3) c

Solving these equations, we find that the sequence follows the rule:

tn n2 - 1

This means the next term when n 6 is:

t6 62 - 1 36 - 1 37

Alternative Methods of Finding the Next Term

In addition to the quadratic sequence approach, there are several alternative methods to derive the next term:

Adding Odd Numbers:

Following the pattern of adding the next odd number to the previous term:

2 3 5 5 5 10 10 7 17 17 9 26 26 11 37

Exponential Pattern:

Another interpretation is to consider the sequence as a step-by-step exponential addition:

2 (base) 5 (2 3) 10 (5 5) 17 (10 7) 26 (17 9) 37 (26 11)

Power of 2:

Another amusing interpretation, as suggested by the book The Hitchhiker's Guide to the Galaxy by Douglas Adams:

2 1^2 - 1 5 2^2 - 1 10 3^2 - 1 17 4^2 - 1 26 5^2 - 1 37 6^2 - 1

Conclusion

The sequence 2, 5, 10, 17, 26, next term is 37. As demonstrated, the quadratic approach provides a straightforward and verifiable method to solve such problems. However, the beauty of such sequences lies in their diversity, allowing multiple interpretations and solutions. This process is not just about getting the right answer but also about understanding the variety of mathematical patterns underlying such sequences.

For more on quadratic sequences and other number patterns, refer to the resources listed below:

Math is Fun: Sequences and Series Khan Academy: Patterns and Sequences