What is Geometric Sequence? | Definition, Properties, Examples, Pros vs Cons

A geometric sequence is also frequently referred to as geometric progression. In the field of mathematics, it is a series of numbers. In this series, each number is followed by another derived by multiplying the previous with a fixed integer (not 1). This number by which it is multiplied is termed the common ratio.

Generally, the constant common ratio is denoted by the letter ‘r,’ whereas the first term of the series is denoted by the letter ‘a’. The formula for arriving at the geometric sequence is thus represented as follows:

a, ar, ar²,ar³, ar⁴….

Key Takeaways

1. A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant factor called the common ratio.
2. In a geometric sequence, the nth term can be found by multiplying the first term by the common ratio raised to the power of n-1.
3. Geometric sequences can be used in various real-world situations, such as calculating compound interest or population growth, and can be modelled by exponential functions.

Example of a geometric sequence

 A simple example of a geometric sequence is the series 2, 6, 18, 54… where the common ratio is 3. Each number is multiplied by 3 to derive the next number in the sequence. Three times two yields 6, which is the second number. Six times three gives 18, which is consequently the following number.

Different properties of a geometric sequence

1. If the common ratio is 1, the sequence becomes constant; the value is the same every time in the series.
2. If the common ratio exceeds 1, the sequence progresses toward infinity. This may be positive or negative, depending upon the sign attached to the first term in the sequence.
3. If the common ratio is positive, all the terms in the sequence will be positive or negative depending on the sign of the initial term. If the common ratio is negative, the signs of the numbers in the series will alternate between positive and negative.
4. If the common ratio is between 1 and -1 (but not 0), then the terms in the series will proportionately tend towards 0.

Advantages of using a geometric sequence

1. The geometric sequence is very useful, particularly in computer programming. This has been used to develop several software and many commonly used apps based on this sequence.
2. A geometric sequence has been known to be used to feed data into machines to generate the easiest way to assemble parts of objects.
3. In other fields of science and mathematics, a geometric sequence may be used to predict future calculations. Since this sequence can be used to derive individual terms up to infinity, this can be used at various points to determine whether the inquiry process will yield desirable results.
4. The knowledge of geometric sequence is a basic necessity for deriving more complex numeric relations, such as geometric progression.

Disadvantages of using a geometric sequence

1. In calculations where the common ratio is not constant, the geometric sequence cannot be used to derive results.
2. Whenever the common ratio has decimal values, the calculations become nearly impossible to simplify beyond a point. The sequence tends to go on till infinity.
3. The basic nature of a geometric sequence has been used to solve several longstanding problems in mathematics. However, the simplicity of the sequence itself dictates that it cannot be used as it is beyond a basic level. Other corollaries may be derived.