Let's start by writing the equation as a geometric series:
1 + x + x^2 + x^3 + ... = 4
We can then factor out a 1 from the left-hand side:
1(1 + x + x^2 + x^3 + ...) = 4
We can then divide both sides by 1 - x to get:
1/(1 - x) = 4
This is an equation of the form a/b = c, where a, b, and c are all real numbers. The solution to this equation is:
x = -b/a
In this case, a = 1, b = 1, and c = 4. Plugging these values into the solution formula, we get:
x = -b/a = -1/1 = -1
However, this is not the solution we are looking for. The series 1 + x + x^2 + x^3 + ... diverges when x = -1. This is because the absolute value of the common ratio (x) is greater than 1.
We need to find a solution that makes the series converge. To do this, we need to find a value of x for which the absolute value of the common ratio is less than 1. The only real number x for which this is true is x = -1/3.
When x = -1/3, the common ratio is x = -1/3. The absolute value of this is less than 1, so the series converges. Therefore, the solution to 1 + x + x^2 + x^3 + ... = 4 is x = -1/3.
