To find the sum of the first six terms of the geometric series given by the numbers 2, 6, 18, and 54, we first need to identify the first term and the common ratio of the series.
The first term (
a) in this series is 2.
To find the common ratio (
r), we divide any term in the series by the term that precedes it. For example:
- The ratio between the second term (6) and the first term (2) is:
- r = 6 / 2 = 3
- The ratio between the third term (18) and the second term (6) is:
- r = 18 / 6 = 3
- The ratio between the fourth term (54) and the third term (18) is:
- r = 54 / 18 = 3
Since the common ratio is consistent throughout, we can confirm that this is indeed a geometric series with a common ratio of 3.
The sum of the first n terms of a geometric series can be calculated using the formula:
Sn = a * (1 – rn) / (1 – r), where:
- Sn is the sum of the first n terms,
- a is the first term,
- r is the common ratio,
- n is the number of terms.
In our case:
- a = 2
- r = 3
- n = 6
Plugging these values into the formula, we get:
S6 = 2 * (1 – 36) / (1 – 3)
First, we calculate 36: 36 = 729.
Now substituting this back in:
S6 = 2 * (1 – 729) / (1 – 3)
S6 = 2 * (-728) / (-2)
This simplifies to:
S6 = 2 * 364 = 728
Therefore, the sum of the first six terms of the geometric series 2, 6, 18, 54 is 728.