91Ó°ÊÓ

We have the sequence as $6,−24,96,−384,1536,−6144,...$

To find the twelfth term and the general term of the sequence, we need to know the common ratio first. So, by comparing the terms, we will find the ratio. After getting the ratio, we will put the value of the first term and the ratio in the general term formula to get the twelfth term and the general term for the sequence.

We have the sequence as $6,−24,96,−384,1536,−6144,...$

The ratio can be calculated as-

1. $\frac{-24}{6}=-4$
   
2. $\frac{96}{-24}=-4$
   
3. $\frac{-384}{96}=-4$
   
4. $\frac{1536}{-384}=-4$
   
5. $\frac{-6144}{1536}=-4$
   

Through his we get the $r$, i.e., common ratio of the sequence as$-4$.

We now have the ratio as $-4$.We know the first term, i.e., $6$.

The general term formula of the geometric sequence is $a_n=a_1{r}^{n-1}$.

Substituting the values gives us,

role="math" localid="1645351728712" $$\begin{gathered} a_n=a_1×{\left(r\right)}^{n-1} \\ {a}_{12}=6×{(-4)}^{12-1} \\ {a}_{12}=6×{(-4)}^{11} \\ {a}_{12}=6×(-4,194,304) \\ {a}_{12}=-25,165,824 \end{gathered}$$
Thus we get the twelfth term as$-25,165,824$.

To find the general term for the sequence, we will substitute the value of the first term, i.e., $a_1=6$and the common ration, i.e., $r=-4$in the general term formula of geometric sequence.

This gives us,

$$\begin{gathered} a_n=a_1{r}^{n-1} \\ {a}_n=6{(-4)}^{n-1} \end{gathered}$$
Thus we get the general term of the sequence as$a_n=6{(-4)}^{n-1}$.