Find the sum of the first six terms of the geometric series that begins with three and has a common ratio of two.
To find the sum of the first six terms of this geometric series, we will use the formula for the sum of the first n terms of a geometric series:
S_n = a(1 - r^n) / (1 - r),
where:
- S_n is the sum of the first n terms,
- a is the first term of the series,
- r is the common ratio of the series,
- n is the number of terms in the series.
In this case, a = 3 (the first term of the series), r = 2 (the common ratio), and n = 6 (we want to find the sum of the first six terms). Plugging these values into the formula, we get:
S_6 = 3(1 - 2^6) / (1 - 2)
S_6 = 3(1 - 64) / (1 - 2)
S_6 = 3(-63) / -1
S_6 = -189 / -1
S_6 = 189
Therefore, the sum of the first six terms of the geometric series is 189.