Find x: A Simple Formula to Solve x + x^2 + x^3 +...+ x^n = 1

[michaelkane]

I need to calculate easy solution for following equation.

x + x^2 + x^3 +...+ x^n = 1

I need a simple formula to calculate value of x that satisfies above equation for any value of n.

This is for accounting/MBA has nothing to do with Physics, but I am sure anyone will be able to help me out.

Thanks.

 

[arildno]

Using the geometric series identity: $x+x^{2}+...+x^{n}=x\frac{x^{n}-1}{x-1}$
we may reformulate your equation to:
$$x^{n+1}-2x+1=0$$
I'm not too sure there exist a nice, general solution of this for arbitrary n.

 

[tacman]

To solve this equation, we can use the formula for the sum of a geometric series:

S = a(1-r^n)/(1-r)

Where S is the sum of the series, a is the first term (in this case, x), r is the common ratio (in this case, x), and n is the number of terms.

In this case, we can set S to be equal to 1, since we want the sum to be equal to 1. Therefore, we have:

1 = x(1-x^n)/(1-x)

Simplifying this equation, we get:

1-x = x-x^(n+1)

Factoring out an x, we get:

1-x = x(1-x^n)

Dividing both sides by (1-x), we get:

1 = x^n

Taking the nth root of both sides, we get:

x = 1^(1/n)

Therefore, the solution for x is:

x = 1^(1/n)

This formula will work for any value of n, giving a simple and easy solution to the equation. I hope this helps with your accounting/MBA problem.