Chapter 7.8, Problem 6STP

Write the explicit and recursive formulas for the geometric sequence $-4,-8,-16,-32,\dots$

To determine

To find : the explicit formula and recursive formula for the geometric sequence .

Answer to Problem 6STP

The explicit formula for the given geometric sequence is

  $a_n=-4{\left(2\right)}^{n-1}$

The recursive formula for the geometric sequence is

  $a_1=-4,a_n=2a_{n-1}$ , for $n\ge 2$

Explanation of Solution

Given sequence: $-4,-8,-16,-32,\mathrm{......}$ .

Calculation:

The first term of the sequence is $a_1=-4$ .

The common ratio r for a geometric sequence is obtained by dividing any term in the sequence by the previous term.

Therefore, common ratio for given geometric sequence is

  $\begin{array}{l}r=\frac{a_2}{a_1}\\ r=\frac{-8}{-4}=2\end{array}$

The explicit formula for the given geometric sequence is

  $\begin{array}{l}{a}_n=a_1{r}^{n-1}\\ a_n=-4{\left(2\right)}^{n-1}\end{array}$

where a_n is nth term of the sequence.

The recursive formula for the given geometric sequence is

  $a_1=-4,a_n=r{a}_{n-1}$ , for $n\ge 2$

  $a_1=-4,a_n=2a_{n-1}$ , for $n\ge 2$