The Formula for 1 + x + x^2 + x^3 + ... + x^n
The formula for the sum of a geometric series is a fundamental concept in mathematics, and it has numerous applications in various fields such as algebra, calculus, and engineering. In this article, we will explore the formula for the sum of a geometric series, which is represented as:
1 + x + x^2 + x^3 + ... + x^n
Derivation of the Formula
The formula for the sum of a geometric series can be derived using the following method:
Let's consider the sum of a geometric series:
S = 1 + x + x^2 + x^3 + ... + x^n
We can multiply both sides of the equation by x to get:
xS = x + x^2 + x^3 + ... + x^(n+1)
Now, let's subtract the original equation from the equation multiplied by x:
(xS - S) = (x + x^2 + x^3 + ... + x^(n+1)) - (1 + x + x^2 + x^3 + ... + x^n)
Simplifying the equation, we get:
(x - 1)S = x^(n+1) - 1
Now, dividing both sides by (x - 1), we get the formula for the sum of a geometric series:
S = (x^(n+1) - 1) / (x - 1)
Applications of the Formula
The formula for the sum of a geometric series has numerous applications in various fields such as:
- Algebra: It is used to solve quadratic equations and to find the roots of polynomials.
- Calculus: It is used to find the area under curves and to solve optimization problems.
- Engineering: It is used to design electronic circuits and to analyze signal processing systems.
Examples
- Example 1: Find the sum of the series 1 + 2 + 4 + 8 + ... + 128.
- Using the formula, we get S = (2^8 - 1) / (2 - 1) = 255.
- Example 2: Find the sum of the series 1 + x + x^2 + ... + x^10.
- Using the formula, we get S = (x^11 - 1) / (x - 1).
In conclusion, the formula for the sum of a geometric series is a powerful tool that has numerous applications in various fields. It is essential to understand the derivation of the formula and its applications to solve problems efficiently.
