Identifying Geometric Sequences- Which of the Following is a Geometric Sequence-

Which of the following is a geometric sequence?

In mathematics, 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. This type of sequence is characterized by its consistent pattern of multiplication, which makes it an essential concept in various fields such as algebra, calculus, and finance. In this article, we will explore some examples and discuss how to identify a geometric sequence among a given set of numbers.

A geometric sequence can be represented as follows:

a, ar, ar^2, ar^3, ar^4, …

Where ‘a’ is the first term, ‘r’ is the common ratio, and the sequence continues indefinitely. The common ratio can be any real number, including fractions, decimals, and even irrational numbers. However, it cannot be zero, as dividing by zero is undefined.

Let’s examine some examples to better understand geometric sequences:

1. 2, 6, 18, 54, 162, …
In this sequence, the common ratio is 3, as each term is three times the previous term.

2. 1/2, 1, 2, 4, 8, …
Here, the common ratio is 2, as each term is twice the previous term.

3. -3, -9, -27, -81, -243, …
In this case, the common ratio is -3, as each term is three times the previous term but with a negative sign.

To determine whether a given sequence is geometric, we need to check if the common ratio remains constant throughout the sequence. This can be done by dividing any term by its preceding term. If the result is the same for all consecutive pairs of terms, then the sequence is geometric.

For example, consider the sequence 4, 12, 36, 108, 324, … To find the common ratio, we can divide the second term by the first term:

12 / 4 = 3

Now, let’s divide the third term by the second term:

36 / 12 = 3

We can continue this process for the remaining terms:

108 / 36 = 3
324 / 108 = 3

Since the common ratio (3) remains constant throughout the sequence, we can conclude that 4, 12, 36, 108, 324, … is a geometric sequence.

In summary, identifying a geometric sequence involves recognizing the consistent pattern of multiplication among the terms and verifying that the common ratio remains constant. By understanding the properties of geometric sequences, we can apply this concept to various real-world scenarios and solve complex mathematical problems.