Is 1/2, 3, and 18 a Geometric or Arithmetic Sequence?

Understanding geometric sequences and arithmetic sequences is crucial for a variety of mathematical applications, including financial calculations, computer algorithms, and data analysis. Here, we'll explore whether the sequence 1/2, 3, 18 is geometric or arithmetic, and how to identify these sequences using their defining characteristics and formulas.

Understanding Sequences

A sequence is a list of numbers arranged in a specific order. Sequences can either be geometric or arithmetic. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant factor known as the common ratio. In an arithmetic sequence, each term after the first is found by adding a constant value to the previous term.

Identifying a Geometric Sequence

In a geometric sequence, the relationship between consecutive terms is given by:

an a(n-1) * r

where an is the nth term of the sequence, a(n-1) is the (n-1)th term, and r is the common ratio.

Let's apply this to the sequence provided: 1/2, 3, 18...

Checking the Common Ratio

To check if the sequence is geometric, we need to determine if there is a common ratio between the terms. We do this by dividing each term by the previous term:

3 / (1/2) 6 18 / 3 6

As we can see, the common ratio (r) is consistently 6. This is a clear indicator that the given sequence is a geometric sequence.

Verifying the Sequence

Let's verify the pattern and calculate the next terms in the sequence:

0 1/72 * 6 1/12

1 1/12 * 6 1/2

2 1/2 * 6 3

3 3 * 6 18

4 18 * 6 108

5 108 * 6 648

6 648 * 6 3888

7 3888 * 6 23328

8 23328 * 6 139968

9 139968 * 6 839808

10 839808 * 6 5038848

and so forth.

Conclusion

Therefore, the sequence 1/2, 3, 18 is indeed a geometric sequence with a common ratio of 6. Understanding the properties of geometric sequences is essential for analyzing and modeling many real-world phenomena, including population growth, compound interest, and radiocarbon dating.

Further Exploration

For more detailed information on geometric and arithmetic sequences, refer to the following resources:

Properties of Geometric Sequences Calculations with Arithmetic Sequences Interpreting Sequences in Data Analysis

Whether you're a student, a professional, or simply interested in expanding your mathematical knowledge, understanding sequences will greatly enhance your analytical skills.