Finding x in a geometric progression, given the sum.

[NotaPhysicist]

Homework Statement

If

$$1 + 2x + 4x^2 + ... = \frac{3}{4}$$

find the value of x. [Edit: Forgot to ask the question]

Homework Equations

$$S_n = \frac{a(1 - r^n)}{1 - r}$$

$$t_n = ar^{n-1}$$

The Attempt at a Solution

a = 1

r = 2x

I try to solve $$S_n$$ and end up with

$$2x^n = \frac{6x - 7}{4}$$

which I can't solve.

I try to solve by equating t2 and t3 and getting x = (1/2). Which is wrong.

Any help appreciated.

 

[zorro]

Is the sum is to infinity?
or to 'n terms'?

If it is to infinity, Apply limit to your equation.

 

[HallsofIvy]

Homework Statement

If

$$1 + 2x + 4x^2 + ... = \frac{3}{4}$$

Homework Equations

$$S_n = \frac{a(1 - r^n)}{1 - r}$$

If this is an infinite sum the formula is
$$S_\infty= \frac{a}{1- r}$$

If it is a finite sum, you would need to know how many terms so that "n" would be an actual integer, not a variable.

The "..." at the end of the sum indicates it is an infinite sum.

$$t_n = ar^{n-1}$$

The Attempt at a Solution

a = 1

r = 2x

I try to solve $$S_n$$ and end up with

$$2x^n = \frac{6x - 7}{4}$$

which I can't solve.

I try to solve by equating t2 and t3 and getting x = (1/2). Which is wrong.

Any help appreciated.

 

[NotaPhysicist]

I've edited the original post. The problem is to find the value of x.

Its not an infinite sum. The only solution I can see is to solve for n in the infinite series and the summation, and try to solve simultaneously.

 

[NotaPhysicist]

The "..." at the end of the sum indicates it is an infinite sum.

I had to read your answer a couple of times. I get it now. Its an infinite sum. No powers to work out.

Thank you so much.