If S1 = 0.7 and S2 = 2.1 in geometric series, what would the sum of the first 12 terms in the series be?

I tried doing this, and I got 1.16 or something :S How exactly do I do this? Please tell me what formula to use. I was using the Sn = a(rn-1)/r-1

To find the sum of the first n terms of a geometric series, you can use the formula:

Sn = a * (1 - r^n) / (1 - r),

where Sn represents the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.

In this case, you were given S1 = 0.7 and S2 = 2.1. Since the sum of the first term is 0.7 and the sum of the second term is 2.1, we can find a and r using these values.

First, we can set up two equations using the formula for Sn. Let's use n = 1 for the first equation and n = 2 for the second equation:

0.7 = a * (1 - r^1) / (1 - r), equation 1
2.1 = a * (1 - r^2) / (1 - r). equation 2

Next, we can solve these equations simultaneously to find the values of a and r. We can do this by expanding the equations and simplifying them:

0.7(1 - r) = a * (1 - r),
2.1(1 - r) = a * (1 - r^2).

Dividing the two equations, we get:

(0.7(1 - r)) / (2.1(1 - r)) = a * (1 - r) / (a * (1 - r^2)),
0.333... = 1 - r / 1 - r^2.

Since we have a quadratic equation, we can solve it by setting it equal to zero:

r^2 - 2r + 0.666... = 0.

Using the quadratic formula, we find:

r = (2 ± √(-1.666...)) / 2.

Since a geometric series has a positive common ratio, we can ignore the negative value in the ± symbol. Therefore, we have:

r = 1 + √(4/3) ≈ 1.1547.

Now that we have the value of r, we can substitute it back into one of the original equations (Equation 1 or 2) to find the value of a. Let's use Equation 1:

0.7 = a * (1 - (1.1547)^1) / (1 - 1.1547),
0.7 = a * (1 - 1.1547) / (1 - 1.1547),
0.7 = -0.1547a.

Simplifying, we find:

a = -0.7 / 0.1547 ≈ -4.523.

Now that we have the values of a and r, we can substitute them into the formula for Sn to find the sum of the first 12 terms:

S12 = (-4.523 * (1 - (1.1547)^12)) / (1 - 1.1547).

Evaluating this expression, we find:

S12 ≈ -4.962.

So, the sum of the first 12 terms in the geometric series is approximately -4.962.

If S1 is supposed to mean the sum of one term, then clearly

a = .7
S2 = a(r^2 - 1)/(r-1)
2.1 = .7(r^2 - 1)/(r-1)
divide by .7
3 = (r^2 - 1)/(r-1)
r^2 - 1 = 3r - 3
r^2 - 3r + 2 = 0
(r-1)(r-2) = 0
r = 1 or r = 2
but if r = 1, the terms would not change and S2 = 1.4 which it is not,
so r = 2
check: .7 + 1.4 = 2.1

S12 = .7(2^12 - 1)/(2-1) = 2866.5

(easier way to find r):
S2 = t1 + t2
2.1 = .7 + t2
t2 = 2.1-.7 = 1.4
t2 = ar
1.4 = .7r
r = 2