Final answer:
To write the geometric series in closed form, use the formula for the sum of a geometric sequence and find the common ratio 'r'. Plug in the values from the given series and simplify to obtain the closed form of the series.
Step-by-step explanation:
To write the given geometric series in closed form using the formula for the sum of a geometric sequence, we need to find the common ratio 'r'. This can be done by dividing any term by its previous term. In this case, 12 divided by 4 gives us 3, so 'r' is equal to 3. The formula for the sum of a geometric sequence is given as:
S = a(1 - r^n) / (1 - r)
where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.
Plugging in the values from the given series, we have:
S = 4(1 - 3^n) / (1 - 3)
Simplifying further, we get:
S = 4(3^n - 1) / 2
Therefore, the closed form of the given geometric series is 4(3^n - 1) / 2.