To find the sum of the geometric sequence with the first three terms of 1, 3, and 9, we need to identify the common ratio and then use the formula for the sum of a geometric series.
The first term of the geometric sequence (commonly denoted as a) is:
- a = 1
The common ratio (denoted as r) can be calculated by dividing the second term by the first term:
- r = 3 / 1 = 3
The third term is:
- 9 = 3 * 3
This confirms our common ratio of 3. The formula for the sum of the first n terms of a geometric sequence is:
Sn = a * (1 – rn) / (1 – r)
Where:
- Sn = sum of the first n terms
- a = first term
- r = common ratio
- n = number of terms
In our case:
- a = 1
- r = 3
- n = 12
Substituting these values into the formula gives:
S12 = 1 * (1 – 312) / (1 – 3)
Calculating 312:
- 312 = 531441
Now substituting this back into our sum formula:
S12 = 1 * (1 – 531441) / (1 – 3)
S12 = (1 – 531441) / (-2)
S12 = -531440 / -2
S12 = 265720
Thus, the sum of the first 12 terms of the geometric sequence is 265720.