Introduction

Determining the nth term formula for a sequence can be a fascinating mathematical challenge. In this article, we will explore the process of finding the nth term for the sequence 7, 13, 23, 37. Understanding the patterns and applying algebraic techniques will enable us to derive a general formula that predicts any term in the sequence.

Identifying the Pattern

To start, let's examine the differences between consecutive terms in the sequence:

13 - 7  623 - 13  1037 - 23  14

Next, we’ll calculate the second differences:

10 - 6  414 - 10  4

Since the second differences are constant and equal to 4, we can conclude that this sequence is quadratic. The general form of a quadratic sequence is An An2 Bn C.

Formulating the Equation

We need to determine the coefficients A, B, and C. We can set up a system of equations based on the known terms of the sequence:

For n 1: A12 B1 C 7 For n 2: A22 B2 C 13 For n 3: A32 B3 C 23

Expressed as:

A1 B C 7 (Equation 1) 4A 2B C 13 (Equation 2) 9A 3B C 23 (Equation 3)

Let's solve this system step by step:

Step 1: Subtract Equation 1 from Equation 2

4A   2B   C - (A   B   C)  13 - 73A   B  6   (Equation 4)

Step 2: Subtract Equation 2 from Equation 3

9A   3B   C - (4A   2B   C)  23 - 135A   B  10   (Equation 5)

Step 3: Subtract Equation 4 from Equation 5

5A   B - (3A   B)  10 - 62A  4A  2

Substitute A 2 back into Equation 4:

3(2)   B  66   B  6B  0

Substitute A 2 and B 0 back into Equation 1:

2   0   C  7C  5

Therefore, the formula for the nth term of the sequence is:

An  2n2   5

Verification

To confirm the accuracy of our formula, let's check the first few terms:

nAn 12(1)2 5 7 22(2)2 5 13 32(3)2 5 23 42(4)2 5 37

The formula produces the correct sequence, verifying its accuracy.

Conclusion

The nth term formula for the sequence 7, 13, 23, 37 is:

boxed{2n^2 5}

This method can be applied to other quadratic sequences by identifying the patterns in first and second differences and using algebraic techniques to derive the general formula. Understanding these steps is essential for solving a wide range of mathematical problems involving sequences.