Geometric sequence puzzler

Question

The following problem has been puzzling me for a couple of days (nb: this is not homework).

There exists two geometric sequences that sum to 9. The value of their second term (t2) is 2.

1. Find the common ratio (r)
2. Find the first element (t1) of each

The answers to (1) are 2/3 and 1/3 and the answers to (2) are 3 and 6 respectively. Unfortunately, I can't understand how these were derived.

In tackling (1) I've tried to apply algebraic substitution to solve for r as follows:

t2 = t1*r; since t2 = 2 we have:
t1 = 2/r

The equation for calculating the sum (S) of a sequence that converges to a limit is given by:

S  = t1 / (1 - r)

So, I tried to plug my value of t1 into S and solve for r as follows:

9 = (2/r) / (1-r)
9(1-r) = 2/r
2/9 = r(1-r)

Unfortunately, from this point I get stuck. I need to eliminate one of the r's but I can't seem to be able to.

Next, I thought to solve for r using the formula that sums the first 2 terms (S2) of the sequence:

S2 = (t1 (1-r^2)) / (1-r)
t1 + 2 = (t1 (1-r^2)) / (1-r)

but expanding this out I again run into the same problem (can't eliminate one of the r's).

So I have 2 questions:

1. What am I doing wrong when deriving r?
2. Once I have one of its values, how I derive the other?

Answer 1

2/9 = r(1-r)

Unfortunately, from this point I get stuck. I need to eliminate one of the r's but I can't seem to be able to.

You need to learn to factorise!

2/9 = r(1-r)
2/9 = r - r^2
2 = 9r - 9r^2
9r^2 - 9r + 2 = 0
(3r)^2 - 3(3r) + 2 = 0

to make it easier, let R = 3r

R^2 - 3R + 2 = 0
(R - 1)(R - 2) = 0

so 3r - 1 = 0, or 3r - 2 = 0
i.e. r = 1/3 or r = 2/3.

And your first term is 2/(1/3) = 6, or 2/(2/3) = 3

QED!

Answer 2

2/9 = r (1 - r)

Rewrite this as ax² + bx + c and use the quadratic formula to solve it:

2/9 = r - r²
r² - r + 2/9 = 0

Using the quadratic formula, the roots are:
[-1 ± √(1 - 8/9)] / 2
= (1 ± 1/3) / 2
= 1/2 ± 1/6
= 1/3 or 2/3

Edit: Aw shoot, I spent way too long figuring out how to write plus/minus and square root. :-P