Understanding Quadratic Sequences: Identifying and Solving the n-th Term

In the field of mathematics and number theory, sequences play a crucial role in understanding patterns and developing analytical skills. One such fascinating sequence is 1 7 17 31... where each term follows a specific pattern. This article delves into the process of identifying the n-th term of this sequence, providing a detailed step-by-step guide for future reference. By the end of this article, readers will not only understand how to solve this particular sequence but also how to apply these methods to other quadratic sequences.

Identifying the Nature of the Sequence

Our sequence is 1 7 17 31... To begin with, it's essential to identify the type of sequence we are dealing with. One way to do this is by examining the first differences between consecutive terms:

7 - 1 6 17 - 7 10 31 - 17 14

Next, we check the differences of these differences:

10 - 6 4 14 - 10 4

Since the second differences are constant (4 in this case), it suggests that the sequence is quadratic in nature.

Formulating the Quadratic Equation

Given a quadratic sequence, the n-th term can be expressed in the form:

a_n An^2 Bn C

We have the first few terms of the sequence:

a_1 1 a_2 7 a_3 17 a_4 31

To find the coefficients A, B, and C, we set up a system of equations based on these terms:

A(1)^2 B(1) C 1 A(2)^2 B(2) C 7 A(3)^2 B(3) C 17 A(4)^2 B(4) C 31

This simplifies to:

A B C 1 (Equation I) 4A 2B C 7 (Equation II) 9A 3B C 17 (Equation III)

Solving the System of Equations

Next, we eliminate C by subtracting Equation I from Equation II and Equation III:

(4A 2B C) - (A B C) 7 - 1 3A B 6 (Equation IV) (9A 3B C) - (4A 2B C) 17 - 7 5A B 10 (Equation V)

Now, we subtract Equation IV from Equation V:

(5A B) - (3A B) 10 - 6

2A 4

A 2

Substituting A 2 into Equation IV:

3(2) B 6

6 B 6

B 0

Now, substituting A 2 and B 0 into Equation I:

2 0 C 1

C -1

Thus, we have:

A 2, B 0, C -1

The n-th term of the sequence is:

a_n 2n^2 - 1

Conclusion

In summary, the n-th term of the sequence 1 7 17 31... is:

2n^2 - 1

Solving quadratic sequences involves identifying patterns, setting up a system of equations, and solving for the unknown coefficients. By understanding and applying these methods, you can efficiently solve a wide range of quadratic sequences.