To show that a series is geometric, you need to demonstrate that the ratio between consecutive terms is constant. This constant ratio is known as the common ratio, usually denoted as r.
Here's a breakdown of how to do it:
1. Understand Geometric Series and Sequences
- A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant r.
- A geometric series is the sum of the terms of a geometric sequence.
2. Calculate the Ratio Between Consecutive Terms
- Take any two consecutive terms in the series. Let's say the terms are an and an+1.
- Calculate the ratio: r = an+1 / an
3. Repeat for Several Pairs of Terms
- Repeat the ratio calculation for several different pairs of consecutive terms in the series. It's crucial to do this for more than just one pair. For example, calculate a2 / a1, a3 / a2, a4 / a3, and so on.
4. Verify the Ratio is Constant
- If the ratio r is the same for all pairs of consecutive terms you tested, then the series is geometric.
Example
Consider the series: 2 + 6 + 18 + 54 + ...
- a1 = 2
- a2 = 6
- a3 = 18
- a4 = 54
Let's calculate the ratios:
- a2 / a1 = 6 / 2 = 3
- a3 / a2 = 18 / 6 = 3
- a4 / a3 = 54 / 18 = 3
Since the ratio is consistently 3, this series is geometric with a common ratio of r = 3.
General Form
A geometric series can be represented in the form:
a + ar + ar2 + ar3 + ...
where:
- a is the first term
- r is the common ratio
In Summary
To prove a series is geometric, calculate the ratio between consecutive terms. If this ratio is constant across multiple pairs of terms, the series is geometric.
