Find the first three terms a geometric sequence

[NotaPhysicist]

Homework Statement

Find the first three terms of a geometric sequence given that the sum of the first four terms is 65/3 and the sum to infinity is 27.

Homework Equations

[tex]
\begin{array}{1}
S_n = \frac{a(1 - r^n)}{1 - r}\\
S_n = \frac{a(r^n - 1}{r - 1}
\end{array}
[/tex]

The Attempt at a Solution

I'm trying to get two equations in two unknowns.

I end with

a + 27r = 27

But I get lost after that. Trying to solve the sum leaves me with a equation to fourth or fifth power which I can't solve.

I'm sure there's a simple approach to this.

 

[danago]

Do you know the formula for the sum to infinity for a geometric series?

http://en.wikipedia.org/wiki/Geometric_progression#Infinite_geometric_series

EDIT: Sorry, just realized that is where you got your equation from.

Consider this:

[tex]\begin{array}{l}
{S_\infty } = \frac{a}{{1 - r}}\\
{S_3} = \frac{{a(1 - {r^3})}}{{1 - r}} = {S_\infty }(1 - {r^3})
\end{array}[/tex]

Do you see how that can help?

 

[Mekanpreth]

is there a latex form to be typed

 

[NotaPhysicist]

For some reason my latex code isn't working. Latex is voodoo magic. No doubt about it.

 

[NotaPhysicist]

I'm still stuck.

So I end up with

[tex] S_3 = 27(1 - r^3) [/tex]

Then where do I go from there? Trying to find the common ratio leaves me with a mess. I know I should end up with a quadratic equation and two sets of solutions, but I completely stumped on how to get there.

 

[Mentallic]

Using the same idea Danago has given, what is S₄?

 

[HallsofIvy]

For some reason my latex code isn't working. Latex is voodoo magic. No doubt about it.

Either that or you used "[\tex]" rather than the correct "[/tex]" to end it! I have corrected it.

 

[NotaPhysicist]

Using the same idea Danago has given, what is S₄?

I got it! Solving for S_4 as above yields a value for r, and from there the other values can be solved, not a quadratic equation in sight.

Thank you. I'm a bit slow on the uptake, but I'm beginning to understand how you guys work now. You don't just hand out the fish, you instead teach us how to fish. Your help is highly appreciated. Thanks again.

 

[Mentallic]

Good on you

Well of course! Your markers are going to determine how well you can fish by showing them there and then what you can do, not what fish you've caught in your homework

I think I took the fish analogy too far hehe... Best of luck Notaphysicist!