Understanding the Common Ratio in Geometric Sequences: Definition and Examples

Common Ratio

In mathematics, specifically in geometric sequences and series, the common ratio refers to the constant ratio between consecutive terms

In mathematics, specifically in geometric sequences and series, the common ratio refers to the constant ratio between consecutive terms.

When we have a geometric sequence, each term is obtained by multiplying the previous term by a fixed number, which is called the common ratio. We can represent a geometric sequence with the formula: a, ar, ar^2, ar^3, …, where ‘a’ is the first term and ‘r’ is the common ratio.

For example, consider the sequence 2, 6, 18, 54, 162,… In this sequence, the common ratio between consecutive terms is 3, because each term is obtained by multiplying the previous term by 3. We can see that 6 = 2 * 3, 18 = 6 * 3, 54 = 18 * 3, and so on.

The common ratio can be positive or negative, depending on whether the terms are increasing or decreasing. If the common ratio is greater than 1, then the terms of the sequence will increase exponentially. If the common ratio is between -1 and 1, then the terms of the sequence will decrease exponentially.

It’s worth noting that if the common ratio is 1, then the sequence will be a constant sequence, where each term is equal to the previous term. In this case, the sequence loses its geometric nature.

To find the common ratio of a geometric sequence, you can divide any term by the previous term. The result will be the common ratio.

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