Chapter 7.8, Problem 25PPE

To determine

Tofind:The explicit formula for thegiven geometric sequence.

Answer to Problem 25PPE

The explicit formula for the geometric sequence $3,-12,48,-192,\dots$ is $a_n=3\cdot {\left(-4\right)}^{n-1}$ .

Explanation of Solution

Given information:

The given geometric sequence is $3,-12,48,-192,\dots$ .

Conceptused:

A geometric sequence is a sequence in which the ratio consecutive terms is constant.

A sequence of the form $a_1,a_{1}r,a_1{r}^2,a_1{r}^3,\dots ,a_1{r}^{n-1},\dots$ is the geometric with $a_1$ as the first term, r the common ratio, and nth term $a_n=a_1{r}^{n-1}$ . This is also known as explicit formula for the sequence.

The geometric sequence may be obtained as $\langle s\rangle =a_1,a_2,a_3,a_4,\dots$

The given geometric sequence is $\langle s\rangle =3,-12,48,-192,\dots$ , that may be written as

  $\langle s\rangle =3,3\times \left(-4\right),3\times {\left(-4\right)}^2,3\times {\left(-4\right)}^3,\dots ,3\times {\left(-4\right)}^{n-1},\dots$ .

The first term of the sequence is 3and the common ratio is $-4$ .

Therefore, the explicit formula for the sequence is $a_n=3\cdot {\left(-4\right)}^{n-1}$ .

Conclusion:

The explicit formula for the geometric sequence $\langle s\rangle =3,-12,48,-192,\dots$ is $a_n=3\cdot {\left(-4\right)}^{n-1}$ .