a geometric series where the first term is - 12, the last term is - 972, and each term after the first is triple the previous term

the geometric series is a (n) = - 12 (3) ^ (n-1)

Step-by-step explanation:

"Triple" denotes multiplication by 3. Thus, the common factor here is 3.

The general formula for a geometric series is a (n) = a (1) (r) ^ (n-1), where a (1) is the first term, r is the common ratio.

Here, we have a (n) = (-12) (3) ^ (n-1) = - 972.

We need to solve this for n, which represents the last term.

The first step towards solving for n is to divide both sides by - 12:

3^ (n-1) = 81

To solve for n-1, rewrite 81 as 3^4. Then we have:

3^ (n-1) = 3^4, implying that (n-1) = 4 and that n = 5.

Then we know that it is the 5th term that equals - 972.

In summary, the geometric series is a (n) = - 12 (3) ^ (n-1).