Find the Next Term 81 , -27 , 9 , -3 , 1 , -1/3

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by ${\displaystyle -\frac{1}{3}}$ gives the next term. In other words, ${\displaystyle a_n=a_1{r}^{n-1}}$.

Geometric Sequence: ${\displaystyle r=-\frac{1}{3}}$

This is the form of a geometric sequence.

Substitute in the values of ${\displaystyle a_1=81}$ and ${\displaystyle r=-\frac{1}{3}}$.

One to any power is one.

Combine ${\displaystyle {(-1)}^{n-1}}$ and ${\displaystyle \frac{1}{3^{n-1}}}$.

Combine ${\displaystyle 81}$ and ${\displaystyle \frac{{(-1)}^{n-1}}{3^{n-1}}}$.

Substitute in the value of ${\displaystyle n}$ to find the ${\displaystyle n}$th term.