Chapter 7.9, Problem 9CYU

To determine

To Find: The $nth$ term and fifth term of geometric sequence $-6,-24,-\mathrm{96.............}$ .

Answer to Problem 9CYU

  $a_n=-6{\left(4\right)}^{n-1}\&a_5=-1536$

Explanation of Solution

Given information:

Geometric sequence is $-6,-24,-\mathrm{96.............}$ .

Concept and Formula Used:

Geometric Sequence: Sequence in which each term after the first is found by multiplying the previous term by a constant called common ratio.

$nth$ term of the geometric sequence is $a_n=a{r}^{n-1}$ where ‘a’ is the first term and ‘r’ is the common ratio.

Calculation:

It is given that geometric sequence is $-7,21,-\mathrm{63.....}$

Common ratio is

  $\begin{array}{l}r=\frac{-24}{-6}=\frac{-96}{-24}=4,a=-6\\ \text{Then }nth\text{ term of the sequence is}\\ a_n=\left(-6\right){\left(4\right)}^{n-1}\\ a_n=-6{\left(4\right)}^{n-1}\end{array}$

Now, the fifth term is

  $\begin{array}{l}a=-6,r=4\&n=5\\ a_5=\left(-6\right){\left(4\right)}^{5-1}\\ a_5=-6{\left(4\right)}^4\\ a_5=-1536\end{array}$

Conclusion:

  $a_n=-6{\left(4\right)}^{n-1}\&a_5=-1536$