How to Find the nth Term of an Arithmetic Sequence

Master the art of identifying patterns and deriving the formula for any term in a given sequence. This guide will step you through the process with practical examples and explanations to enhance your SEO for Google's standards.

Introduction

Sequences are fundamental in mathematics, and understanding how to find the nth term is a crucial skill. In this article, we will explore how to derive the formula for the nth term of a specific sequence, 5 10 20 40 80. We'll break down the steps, highlight the significance of each step, and ensure our content is optimized for Google's search engine standards.

Analyzing the Sequence

The sequence 5 10 20 40 80 might seem random at first glance, but it's easier to identify the pattern once we examine its structure. Let's proceed step by step.

Identifying the Pattern

Let's write down the sequence and identify the relationship between each term:

The first term a1 is 5. The second term a2 is 10, which is 5 multiplied by 2. The third term a3 is 20, which is 10 multiplied by 2. The fourth term a4 is 40, which is 20 multiplied by 2. The fifth term a5 is 80, which is 40 multiplied by 2.

From this analysis, we can observe a consistent pattern: each term is twice the previous term. This indicates that the sequence is a geometric progression with a common ratio of 2.

General Formula for the nth Term

To find the nth term, we use the formula for a geometric progression:

an a1 × r(n-1)

Where:

a1 is the first term, r is the common ratio, and n is the term number.

Substitute the values for the given sequence:

an 5 × 2(n-1)

Using this formula, we can verify the sequence:

(a1#32;#32;5 × 2(1-1) 5 × 20 5) (a2#32;#32;5 × 2(2-1) 5 × 21 10) (a3#32;#32;5 × 2(3-1) 5 × 22 20) (a4#32;#32;5 × 2(4-1) 5 × 23 40) (a5#32;#32;5 × 2(5-1) 5 × 24 80)

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

an 5 × 2(n-1)

Geometric Progression and nth Term Formula

Geometric Progression (GP) is a sequence where each term is obtained by multiplying the previous term by a constant, known as the common ratio. The formula for the nth term of a GP is:

Tn a × r(n-1)

Where:

a is the first term, r is the common ratio, and n is the term number.

Practice Example with Negative Sequence

Let's take another sequence -5 -10 -20 -40 -80. This is also a geometric sequence with a common ratio of 2 and a first term of -5:

Tn -5 × 2(n-1)

We can verify the sequence using the formula:

(T1#32;#32;-5 × 2(1-1) -5 × 20 -5) (T2#32;#32;-5 × 2(2-1) -5 × 21 -10) (T3#32;#32;-5 × 2(3-1) -5 × 22 -20) (T4#32;#32;-5 × 2(4-1) -5 × 23 -40) (T5#32;#32;-5 × 2(5-1) -5 × 24 -80)

Conclusion

Understanding the pattern in a sequence and deriving the formula for the nth term is a valuable skill in mathematics. By following the steps outlined, you can confidently solve similar problems and enhance your SEO for Google's standards. Remember to practice with various examples to deepen your understanding and proficiency.