Geometric sequences question, don't know where to start!

[gymnastqueen]

The question I was given is: "Determine two geometric sequences whose first three terms are 18x-9, 2x+8, and x-1. (Hint: You will find using the quadratic formula helpful.)". I've never been given a question like and am really unsure where to start. As well I'm not very good at anything having to do with the quadratic formula, but I'm pretty decent at working with sequences so if anybody has any ideas on how to get started on this one it would be much appreciated! Thanks.

 

[skeeter]

geometric sequences have a common ratio ...

\(\displaystyle \frac{2x+8}{18x-9} = \frac{x-1}{2x+8}\)

can you work from here?

 

[gymnastqueen]

This helps a bit, at least I have a starting point. What you gave me should help me find the common ratio right? Any idea how I would link this to the quadratic formula to find the sequences? Sorry, I'm really bad at quadratic stuff. As well because they're asking for two sequences would you say that one is most likely explicit and the other recursive, or does that matter? I've been trying to use t[sub:1oq3zz1k]n[/sub:1oq3zz1k]= ar[sup:1oq3zz1k]n-1[/sup:1oq3zz1k] is this a step in the right direction? Thanks

 

[Mrspi]

This helps a bit, at least I have a starting point. What you gave me should help me find the common ratio right? Any idea how I would link this to the quadratic formula to find the sequences? Sorry, I'm really bad at quadratic stuff. As well because they're asking for two sequences would you say that one is most likely explicit and the other recursive, or does that matter? I've been trying to use t[sub:1lxfyfmj]n[/sub:1lxfyfmj]= ar[sup:1lxfyfmj]n-1[/sup:1lxfyfmj] is this a step in the right direction? Thanks

You could cross-multiply, which is the first step that many would use to solve a proportion:

(2x + 8)(2x + 8) = (18x - 9)(x - 1)

Now, do the multiplications....and you should find that you have a quadratic equation. Get 0 on one side, and then apply the quadratic formula (or, maybe you can factor). This should give you two values for x.

Once you have x, you can use each of the values to write out a few terms of a sequence (using the expressions given for the terms). You'll have the first term, or "a", and you can find the value of r for each sequence...then use your formula

t[sub:1lxfyfmj]n[/sub:1lxfyfmj] = a * r[sup:1lxfyfmj]n - 1[/sup:1lxfyfmj]

Since you have two different values for a, and two different values for r, you can write formulas for two different sequences.