A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In other words, it is a series where each term is obtained by multiplying the previous term by a constant ratio.
The concept of geometric series involves several key points:
The formula for the sum of a geometric series is given by:
Where:
To apply the geometric series formula, follow these steps:
The symbol commonly used to represent a geometric series is the capital letter "S" with a subscript indicating the number of terms. For example, Sₙ represents the sum of a geometric series with n terms.
There are several methods to solve problems involving geometric series:
Find the sum of the geometric series 2, 4, 8, 16, 32, ... with 6 terms.
Determine the sum of the geometric series 3, 6, 12, 24, ... with an infinite number of terms.
Q: What is a geometric series? A: A geometric series is a sequence of numbers in which each term is obtained by multiplying the previous term by a constant ratio.
Q: How do you find the sum of a geometric series? A: The sum of a geometric series can be found using the formula: , where a is the first term, r is the common ratio, and n is the number of terms.
Q: Can a geometric series have an infinite sum? A: Yes, a geometric series can have an infinite sum if the absolute value of the common ratio is less than 1.
Q: How can geometric series be applied in real-life situations? A: Geometric series can be used to model exponential growth or decay, such as population growth, compound interest, or radioactive decay.
Q: Are there any other methods to find the sum of a geometric series? A: Besides using the formula, recursive approaches or geometric interpretations can be employed to find the sum of a geometric series.