Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric sequence is:

a₁, a₁ * r, a₁ * r², a₁ * r³, ...

Where:

• a₁ is the first term
• r is the common ratio

The formula to find the nth term of a geometric sequence is:

a_n = a₁ * r^(n-1)

The formula to find the sum of the first n terms of a geometric sequence (also known as the geometric series) is:

S_n = a₁ * (1 - rⁿ) / (1 - r)

Where:

• S_n is the sum of the first n terms
• n is the number of terms

Geometric sequences and series are used in various real-life applications, such as population growth, interest calculations, and exponential decay.

Understanding geometric sequences and series is important in mathematics and can be useful in solving many problems involving exponential growth or decay.