# A geometric sequence is defined by the explicit formula a_n = 5(-3)_(n-1), what is the recursive formula for the nth term of this sequence?

Sep 7, 2016

The recursive formula is:

$\left\{\begin{matrix}{a}_{1} = 5 \\ {a}_{n + 1} = - 3 {a}_{n}\end{matrix}\right.$

#### Explanation:

I assume the formula is: ${a}_{n} = 5 {\left(- 3\right)}^{n - 1}$.

To calculate the recursive formula first we can calculate some terms of the sequence:

${a}_{1} = 5 \cdot {\left(- 3\right)}^{1 - 1} = 5 \cdot {\left(- 3\right)}^{0} = 5$

${a}_{2} = 5 \cdot {\left(- 3\right)}^{2 - 1} = 5 \cdot {\left(- 3\right)}^{1} = - 15$

${a}_{3} = 5 \cdot {\left(- 3\right)}^{3 - 1} = 5 \cdot {\left(- 3\right)}^{2} = 45$

From these calculations we see that the first term is ${a}_{1} = 5$ and each other term comes from multiplying previous one by $- 3$.

This algorithm can be written as the recursive formula:

$\left\{\begin{matrix}{a}_{1} = 5 \\ {a}_{n + 1} = - 3 {a}_{n}\end{matrix}\right.$