Geometric Sequence

What is a Geometric Sequence?

A geometric sequence is a list of numbers where each number is found by multiplying the previous one by the same number. For example, in $2, 4, 8, 16$ , each number is multiplied by $2$ .

General Form of a Geometric Sequence

A geometric sequence can be written as: $a, \, ar, \, ar^2, \, ar^3, \, \dots$

where:

$a$ is the first term of the sequence,
$r$ is the common ratio, and
each term is obtained by multiplying the previous term by $r$ .

For example, in the sequence $2, 6, 18, 54, \dots$ , the first term is $a = 2$ , and the common ratio is $r = 3$ because each term is multiplied by $3$ to get the next term.

Real-World Applications of Geometric Sequences

Finance:
- Compound interest calculations involve geometric sequences.
- Example: If you invest $100$ at an annual interest rate of $10%$ , the amount after $n$ years is a geometric sequence: $100, 110, 121, 133.1, \dots$ .
Science:
- Modeling population growth, radioactive decay, and spread of diseases, which follow geometric patterns.
Technology:
- Signal strength in communication systems often decreases geometrically with distance.