# How do you write the expression for the nth term of the geometric sequence a_4=-18, a_7=2/3, n=6?

Mar 5, 2017

${a}_{n} = 486 {\left(- \frac{1}{3}\right)}^{n - 1}$

#### Explanation:

$\text{The nth term for a geometric sequence is}$

• a_n=ar^(n-1)" where a is the 1st term"

To obtain the nth term for the given sequence, we require to find a and r.

$\text{Given " a_4=-18" and "a_7=2/3" then}$

$\Rightarrow {a}_{4} = a {r}^{3} = - 18 \to \left(1\right)$

$\Rightarrow {a}_{7} = a {r}^{6} = \frac{2}{3} \to \left(2\right)$

$\Rightarrow \frac{a {r}^{6}}{a {r}^{3}} = \frac{\frac{2}{3}}{- 18}$

$\Rightarrow {r}^{3} = - \frac{1}{27} \Rightarrow \textcolor{red}{r = - \frac{1}{3}}$

$\text{From (1) } a {r}^{3} = - 18$

$\Rightarrow a = \frac{- 18}{- \frac{1}{27}} \Rightarrow \textcolor{red}{a = 486}$

$\Rightarrow \text{nth term expression is } {a}_{n} = 486 {\left(- \frac{1}{3}\right)}^{n - 1}$