Answer:
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).
![\ \ \ \ \ \ \ \ \ \Downarrow \ \Downarrow \Downarrow\ \ \ \ \Downarrow \ \ \ \ \Downarrow \\ \\ \left[\begin{array}{ccc} \left[\begin{array}{ccc} \left[\begin{array}{ccc}5(4x + 2)/7y\end{array}\right] \end{array}\right] \end{array}\right]](https://tex.z-dn.net/?f=%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5CDownarrow%20%5C%20%5CDownarrow%20%5CDownarrow%5C%20%5C%20%5C%20%5C%20%5CDownarrow%20%5C%20%5C%20%5C%20%20%5C%20%5CDownarrow%0A%0A%5C%5C%20%5C%5C%20%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D5%284x%20%2B%202%29%2F7y%5Cend%7Barray%7D%5Cright%5D%20%5Cend%7Barray%7D%5Cright%5D%20%5Cend%7Barray%7D%5Cright%5D%20)
