Understanding the Geometric Sequence Common Ratio: A Detailed Guide

Mathematics is a vast universe filled with fascinating concepts that help us understand patterns and relationships. One of these intriguing concepts is the geometric sequence, a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this article, we will explore the common ratio with a particular example. We will first define a geometric sequence, explain how to find the common ratio, and then dive into the specific example given in the question: the sequence 120, 60, 30, 15, 15/2. By the end, you will have a clear understanding of how to determine the common ratio in a geometric sequence.

What is a Geometric Sequence?

A geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant factor. This constant factor is known as the common ratio (denoted as ( r )).

How to Find the Common Ratio in a Geometric Sequence

The common ratio of a geometric sequence can be found by dividing any term in the sequence by the term immediately preceding it. Mathematically, it is expressed as:

( r frac{a_{n 1}}{a_n} )

Where ( a_n ) represents the nth term in the sequence, and ( a_{n 1} ) represents the term following it.

Example: Finding the Common Ratio of the Sequence 120, 60, 30, 15, 15/2

Let's take the example sequence provided: 120, 60, 30, 15, 15/2. We will find the common ratio by dividing each term by its preceding term. We will use the following steps:

Step 1: Divide First Term by Second Term

The first term ( a_1 ) is 120, and the second term ( a_2 ) is 60. Dividing ( a_2 ) by ( a_1 ):

( r frac{a_2}{a_1} frac{60}{120} frac{1}{2} )

Step 2: Divide Second Term by Third Term

Next, we divide the second term ( a_2 ) (which is 60) by the third term ( a_3 ) (which is 30):

( r frac{a_3}{a_2} frac{30}{60} frac{1}{2} )

Step 3: Divide Third Term by Fourth Term

Continue with the third term ( a_3 ) (which is 30) and the fourth term ( a_4 ) (which is 15):

( r frac{a_4}{a_3} frac{15}{30} frac{1}{2} )

Step 4: Divide Fourth Term by Fifth Term

Finally, divide the fourth term ( a_4 ) (which is 15) by the fifth term ( a_5 ) (which is 15/2):

( r frac{a_5}{a_4} frac{15/2}{15} frac{1}{2} )

From all these calculations, we can see that the common ratio ( r ) is consistently 1/2. This confirms that the sequence has a common ratio of 1/2.

Conclusion

Understanding the common ratio in a geometric sequence is fundamental to grasping the nature and behavior of such sequences. In the given example, we found that the common ratio of the sequence 120, 60, 30, 15, 15/2 is 1/2. By following the steps outlined above, you can easily determine the common ratio of any geometric sequence, which is the multiplicative factor between consecutive terms. This basic concept is crucial in a wide range of mathematical applications, from finance to engineering, making it a valuable skill to master.

Further Reading

To delve deeper into the world of geometric sequences and ratios, you may want to explore related topics such as:

The formula for finding the nth term of a geometric sequence The sum of the first n terms of a geometric sequence

By expanding your knowledge in these areas, you can enhance your understanding of how geometric sequences function and apply this knowledge to solve more complex mathematical problems.