Identifying Geometric Sequences- A Comprehensive Guide to Recognizing and Analyzing Them
Which sequences are geometric? This question often arises in mathematics, particularly when dealing with arithmetic progressions and geometric progressions. Understanding the characteristics of geometric sequences is crucial for various mathematical applications and problem-solving. In this article, we will explore the properties of geometric sequences and provide a list of all the applicable sequences that meet the criteria.
A geometric sequence is a sequence 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. The general form of a geometric sequence is:
a, ar, ar^2, ar^3, …
where ‘a’ is the first term, and ‘r’ is the common ratio.
To determine which sequences are geometric, we must check if the ratio between consecutive terms remains constant. Let’s analyze some examples:
1. 2, 4, 8, 16, 32, …
This sequence is geometric because the common ratio is 2 (4/2 = 8/4 = 16/8 = 32/16 = 2).
2. 3, 6, 12, 24, 48, …
Similarly, this sequence is geometric with a common ratio of 2 (6/3 = 12/6 = 24/12 = 48/24 = 2).
3. 5, 10, 20, 40, 80, …
Here, the common ratio is 2 as well (10/5 = 20/10 = 40/20 = 80/40 = 2).
4. 1, 1/2, 1/4, 1/8, 1/16, …
This sequence is also geometric, with a common ratio of 1/2 (1/2/1 = 1/4/1/2 = 1/8/1/4 = 1/16/1/8 = 1/2).
On the other hand, the following sequences are not geometric:
1. 1, 3, 6, 10, 15, …
This sequence is arithmetic, not geometric, as the difference between consecutive terms increases by 1 (3-1 = 2, 6-3 = 3, 10-6 = 4, 15-10 = 5).
2. 2, 4, 8, 12, 16, …
Although the ratio between consecutive terms is 2, the sequence is not geometric because the terms are not obtained by multiplying the previous term by a constant ratio (4/2 = 2, 8/4 = 2, 12/8 = 1.5, 16/12 = 1.333…).
In conclusion, to determine which sequences are geometric, check if the ratio between consecutive terms remains constant. The sequences mentioned above (2, 4, 8, 16, 32, …; 3, 6, 12, 24, 48, …; 5, 10, 20, 40, 80, …; 1, 1/2, 1/4, 1/8, 1/16, …) are geometric sequences, while the others (1, 3, 6, 10, 15, …; 2, 4, 8, 12, 16, …) are not.
