Understanding the Next Number in the Sequence 1 3 7 13 21 31

In this article, we will explore the sequence 1 3 7 13 21 31 and determine the next number in the sequence. We will also discuss the methods and reasoning behind the solution, including identifying the pattern and formulating the sequence to a mathematical rule.

Pattern Identification

The sequence 1 3 7 13 21 31 can be approached in several ways, but one of the simplest and most intuitive methods is to examine the differences between consecutive terms. Let's begin by identifying the differences:

Differences: 2 4 6 8 10

Here, we can see that the differences themselves form an arithmetic sequence with a common difference of 2. This suggests that the sequence we are dealing with is a quadratic sequence, where each term can be expressed as a quadratic polynomial.

Quadratic Sequence Analysis

A quadratic sequence is a sequence where the nth term follows a polynomial of degree 2. The general form for the nth term of a quadratic sequence is:

$$ t_n an^2 bn c $$

To determine the coefficients a, b, and c, we can use the first few terms of the sequence. Let's consider the first three terms:

For ( n 1 ): ( t_1 a(1)^2 b(1) c 1 ) For ( n 2 ): ( t_2 a(2)^2 b(2) c 3 ) For ( n 3 ): ( t_3 a(3)^2 b(3) c 7 )

These can be written as a system of linear equations:

Equation 1: ( a b c 1 ) Equation 2: ( 4a 2b c 3 ) Equation 3: ( 9a 3b c 7 )

By solving this system, we can find the values of a, b, and c.

Solving the System:

Subtract Equation 1 from Equation 2: ( 3a b 2 ) Subtract Equation 2 from Equation 3: ( 5a b 4 ) Solve the resulting system of two equations: Equation 4: ( 3a b 2 ) Equation 5: ( 5a b 4 ) Subtract Equation 4 from Equation 5: ( 2a 2 ) ( a 1 ) Solve for b: ( b 2 - 3a -1 ) Solve for c: ( c 1 - a - b 1 - 1 1 1 )

The coefficients are ( a 1 ), ( b -1 ), and ( c 1 ).

Formulating the Sequence

Now that we have the coefficients, we can express the general term of the sequence as:

$$ t_n n^2 - n 1 $$

To find the next term in the sequence, we substitute ( n 7 ) into the equation:

$$ t_7 7^2 - 7 1 49 - 7 1 43 $$

Verification

To verify, we can check the difference between consecutive terms and ensure the pattern holds:

Differences: 2 4 6 8 10 12 14

The next difference is 12, and adding 12 to 31:

$$ 31 12 43 $$

This confirms that the next term is indeed 43.

Conclusion

The next number in the sequence 1 3 7 13 21 31 is 43. This conclusion is based on identifying the pattern, solving for the coefficients of a quadratic polynomial, and verifying the result.

Keywords: sequence, prime numbers, quadratic sequences

Further Resources

If you are interested in learning more about quadratic sequences and their properties, you can explore the following resources:

"What is a Quadratic Sequence?" on Math Is Fun (link) "Understanding Quadratic Equations" by Khan Academy (link)

Good luck with your sequence analysis and problem solving!