The Formula for 1 + x + x^2 + x^3 + ... + x^n
The formula for the sum of a geometric series, 1 + x + x^2 + x^3 + ... + x^n, is a fundamental concept in mathematics. In this article, we will prove the formula for this sum.
The Formula
The formula for the sum of a geometric series is:
1 + x + x^2 + x^3 + ... + x^n = (x^(n+1) - 1) / (x - 1)
Proof
To prove this formula, we will use the method of mathematical induction.
Base Case
First, we will show that the formula is true for n = 0.
1 + x^0 = 1 = (x^(0+1) - 1) / (x - 1) = (x - 1) / (x - 1) = 1
This shows that the formula is true for n = 0.
Inductive Step
Now, we will assume that the formula is true for n = k. That is, we assume that:
1 + x + x^2 + x^3 + ... + x^k = (x^(k+1) - 1) / (x - 1)
We will show that if the formula is true for n = k, then it is also true for n = k + 1.
1 + x + x^2 + x^3 + ... + x^(k+1) = (1 + x + x^2 + x^3 + ... + x^k) + x^(k+1)
Using the assumption that the formula is true for n = k, we can rewrite the right-hand side as:
= ((x^(k+1) - 1) / (x - 1)) + x^(k+1)
= (x^(k+1) - 1 + x^(k+1) - x^(k+1) + 1) / (x - 1)
= (x^(k+2) - 1) / (x - 1)
This shows that the formula is true for n = k + 1.
Conclusion
By mathematical induction, we have shown that the formula:
1 + x + x^2 + x^3 + ... + x^n = (x^(n+1) - 1) / (x - 1)
is true for all positive integers n.
