A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same constant value, known as the common ratio (r). The sequence grows if the ratio is greater than 1, shrinks if the ratio is between 0 and 1, and alternates in sign if the ratio is negative. If the ratio is exactly 1, the sequence remains constant.
The image below shows how each term in a geometric sequence is calculated by multiplying the first term, labeled a, by powers of the common ratio r. The second term is a × r, the third is a × r², and the nth term is a × rⁿ⁻¹. Each step increases the exponent on r by one.
For example, the sequence 5, 10, 20, 40, 80 is a geometric sequence with a common ratio of 2. Each term is double the one before it. The sequence 100, 50, 25, 12.5… is also geometric, this time with a common ratio of 0.5. And a sequence like 3, –3, 3, –3, … has a common ratio of –1, causing the terms to alternate in sign.
Geometric sequences are often used to model repeated multiplication or exponential change. They appear in contexts like population growth, radioactive decay, financial investments, and the spread of viruses—any situation where the next value depends on a fixed percentage or factor of the previous one.
It’s important to distinguish a geometric sequence from a geometric series. A geometric sequence is just the list of numbers that follow the multiplication rule. A geometric series is the sum of those numbers, which you can find the value by using the geometric sum formula.
Suppose a scientist is studying the decay of a chemical that loses 30% of its mass every hour. If the initial mass is 80 grams, the amount remaining each hour forms a geometric sequence with a common ratio of 0.7: 80, 56, 39.2, 27.44, and so on. This predictable pattern allows researchers to model how much of the substance remains at any point in time.
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