How To Find the Next Number in a Sequence: The Case of 1 3 7 13 21 31 43

Introduction

Sequence puzzles have always been a popular way of entertaining the mind and challenging the analytical skills. One such intriguing sequence is 1, 3, 7, 13, 21, 31, 43. In this article, we will explore various methods to determine the next number in this sequence and discuss the underlying patterns and mathematical concepts involved.

Understanding the Sequence Pattern

First Differences

Let's start by examining the first differences between consecutive terms in the sequence:

We notice that the first differences (2, 4, 6, 8, 10) themselves form an arithmetic sequence with a common difference of 2. Extending this pattern, the next first difference would be 10 2 12.

Therefore, the next term of the original sequence is obtained by adding 12 to the last term (31):

31 12 43

Developing a Formula for the Nth Term

Second Differences and Quadratic Patterns

Given that the first differences are increasing by a constant amount (2), we suspect that the sequence might be quadratic. To confirm this, let's calculate the second differences:

The second differences are constant, indicating that the sequence can be expressed as a quadratic function of the form:

tn an2 bn c

Let's find the values of a, b, c using the first three terms:

1 a(1)2 b(1) c
3 a(2)2 b(2) c
7 a(3)2 b(3) c

Setting up the system of equations:

a b c 1
4a 2b c 3
9a 3b c 7

Solving this system, we get:

a 1, b -1, c 1

Thus, the nth term formula for the sequence is:

tn n2 - n 1

Substituting n 7 to find the 7th term:

t7 72 - 7 1 49 - 7 1 43

Conclusion

In conclusion, there are multiple ways to solve for the next number in a sequence. Whether through first differences, second differences, or developing a quadratic formula, the key is to identify the underlying pattern. For the sequence 1, 3, 7, 13, 21, 31, 43, the next term is indeed 43.

Whether you're a math enthusiast, a puzzle solver, or simply someone curious about patterns and sequences, understanding the methods discussed here can greatly enhance your problem-solving skills.