## geometric sequence

### A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio (r)

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio (r). In other words, each term in the sequence is obtained by multiplying the previous term by the same fixed value.

The general form of a geometric sequence is: a, ar, ar^2, ar^3, ar^4, …

In this sequence, ‘a’ represents the first term of the sequence, and ‘r’ represents the common ratio. The subsequent terms in the sequence are obtained by multiplying the previous term by ‘r’.

For example, let’s consider a geometric sequence starting with the first term 2 and a common ratio of 3. The sequence would be: 2, 6, 18, 54, 162, …

To find a specific term in a geometric sequence, we can use the formula:

Tn = a * r^(n-1)

Where Tn represents the desired term in the sequence, ‘a’ is the first term, ‘r’ is the common ratio, and ‘n’ is the position of the term in the sequence.

For instance, if we want to find the 5th term in the sequence mentioned earlier (2, 6, 18, 54, 162, …), we can use the formula:

T5 = 2 * 3^(5-1) = 2 * 3^4 = 2 * 81 = 162

So, the 5th term in the sequence is 162.

Geometric sequences have various applications in mathematics, science, finance, and other fields where growth or decay follows a consistent ratio.

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