Understanding Geometric Sequences | Definition, Formula, and Examples

Geometric Sequences

A geometric sequence, also known as a geometric progression, is a sequence of numbers in which each term after the first is found by multiplying the preceding term by a fixed, non-zero number called the common ratio

A geometric sequence, also known as a geometric progression, is a sequence of numbers in which each term after the first is found by multiplying the preceding term by a fixed, non-zero number called the common ratio.

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

In this sequence:
– “a” represents the first term,
– “r” represents the common ratio.

For example, let’s consider the geometric sequence: 2, 6, 18, 54, …
Here, the first term (a) is 2, and the common ratio (r) is 3. Each term after the first is obtained by multiplying the preceding term by 3.

To find any term in a geometric sequence, you can use the formula:
Term(n) = a * r^(n-1)
where “n” represents the position of the term you want to find.

Let’s say we want to find the 6th term of the sequence mentioned above:
Term(6) = 2 * 3^(6-1) = 2 * 3^5 = 2 * 243 = 486

In addition to finding specific terms, you can also find the sum of a geometric sequence if it has a finite number of terms. The formula to calculate the sum of a geometric sequence is:
Sum(n) = (a * (r^n – 1))/(r – 1)

For example, if we want to find the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54, …
Sum(5) = (2 * (3^5 – 1))/(3 – 1) = (2 * 242)/2 = 242

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