To find the sum of the first seven terms of the given geometric series, we first need to identify the first term and the common ratio.
The first term (a) is 3. To find the common ratio (r), we can divide the second term by the first term:
r = 12 / 3 = 4.
Now that we have both the first term and the common ratio, we can use the formula for the sum of the first n terms of a geometric series, which is:
S_n = a * (1 – r^n) / (1 – r),
where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
For our case:
- a = 3
- r = 4
- n = 7
Now, we can substitute these values into the formula:
S_7 = 3 * (1 – 4^7) / (1 – 4).
Calculating 4^7 gives us 16384. Now plug that back into the equation:
S_7 = 3 * (1 – 16384) / (1 – 4) = 3 * (-16383) / (-3) = 3 * 5461 = 16383.
So, the sum of the first seven terms of the geometric series is 16383.