If S1 = 0.7 and S2 = 2.1 in a geometric series, determine the sum of the first 12 terms in the series. Be sure to show all of your work.
a = 0.7
a+ar = 2.1
so,
0.7(1+r) = 2.1
1+r = 3
r = 2
So, now you know that
S12 = a(1-r^12)/(1-r)
Just plug in your values.
To determine the sum of the first 12 terms in the geometric series, we can use the formula for the sum of a geometric series:
Sn = a(1 - r^n) / (1 - r)
Where:
Sn = sum of the first n terms
a = the first term in the series
r = the common ratio
n = number of terms
Given:
S1 = 0.7
S2 = 2.1
We can find the common ratio (r) by dividing S2 by S1:
r = S2 / S1
r = 2.1 / 0.7
r = 3
Now we have the first term (a = S1) and the common ratio (r = 3). We need to find the sum of the first 12 terms (n = 12).
Plugging the values into the formula:
Sn = a(1 - r^n) / (1 - r)
S12 = 0.7(1 - 3^12) / (1 - 3)
S12 = 0.7(1 - 531441) / -2
S12 = 0.7(-531440) / -2
S12 = -371008
Therefore, the sum of the first 12 terms in the series is -371008.
To determine the sum of the first 12 terms in a geometric series, we will make use of the formula for the sum of a geometric series:
S = a * (1 - r^n) / (1 - r)
Where:
- S is the sum of the series
- a is the first term in the series
- r is the common ratio
- n is the number of terms in the series
In this case, we are given that S1 = 0.7 and S2 = 2.1. We can use these values to find a and r.
The first term, a, is simply S1, so a = 0.7.
The common ratio, r, can be found by dividing S2 by S1:
r = S2 / S1 = 2.1 / 0.7 = 3
Now we have a = 0.7 and r = 3.
Using the formula for the sum of a geometric series with n = 12, we can find the sum:
S = a * (1 - r^n) / (1 - r)
= 0.7 * (1 - 3^12) / (1 - 3)
= 0.7 * (1 - 531441) / (1 - 3)
= 0.7 * (-531440) / (-2)
= 0.7 * 265720
= 185004
Therefore, the sum of the first 12 terms in the geometric series is 185004.