Geometric Sequences: A Step-by-Step Guide

In the world of mathematics, sequences are like ordered lists of numbers that follow specific patterns. Among these sequences, geometric sequences hold a special place. They're characterized by a constant ratio between consecutive terms, making them predictable and intriguing to explore.

Understanding Geometric Sequences

Imagine a sequence where each number is obtained by multiplying the previous number by a fixed value. This fixed value is known as the common ratio. For instance, consider the sequence: 2, 4, 8, 16, 32... Notice how each term is twice the previous one. Here, the common ratio is 2.

Formally, a geometric sequence can be represented as:

a, ar, ar², ar³, ...

• a: The first term of the sequence.
• r: The common ratio.
• n: The position of a term in the sequence (e.g., the 3rd term, the 5th term, etc.).

The Formula for the nth Term

To find any term in a geometric sequence, we use a simple formula:

a_n = a * r^(n-1)

Where:

• a_n: The nth term of the sequence.
• a: The first term.
• r: The common ratio.
• n: The position of the term.

Examples: Putting the Formula to Work

Example 1: Finding the 6th Term

Let's say we have a geometric sequence with the first term (a) being 3 and a common ratio (r) of 4. We want to find the 6th term (a₆).

Using the formula:

a₆ = 3 * 4^(6-1)

a₆ = 3 * 4⁵

a₆ = 3 * 1024

a₆ = 3072

Therefore, the 6th term of this geometric sequence is 3072.

Example 2: Finding the Common Ratio

Suppose we have a geometric sequence with the first term (a) being 10 and the 3rd term (a₃) being 90. Our goal is to find the common ratio (r).

We know a₃ = a * r^(3-1).

Substituting the values: 90 = 10 * r²

Dividing both sides by 10: 9 = r²

Taking the square root of both sides: r = ±3

Therefore, the common ratio (r) can be either 3 or -3.

Example 3: Real-World Application

Imagine a bank account that earns 5% interest annually. If you initially deposit $1000, the amount in the account will form a geometric sequence. The first term (a) is $1000, and the common ratio (r) is 1.05 (representing a 5% increase). You can use the formula to calculate the balance after any number of years.

Conclusion

Geometric sequences offer a fascinating way to understand patterns and growth. The formula provides a powerful tool for calculating any term in the sequence, making it applicable in various fields like finance, physics, and computer science. By understanding the concept and the formula, you can navigate through the world of geometric sequences with confidence.