Chapter 11.3, Problem 1OE

To determine

To calculate:

Formula for nth term of geometric sequence.

Answer to Problem 1OE

The formula for nth term of geometric sequence is $t_n=5000{\left(\frac{1}{5}\right)}^n$ .

Explanation of Solution

Given information:

Sequence:

  $1000,200,40,8,\mathrm{...}$

Formula used:

  $t_n=t_1{r}^{n-1}$

Where,

  $t_1$ = first number

  $r=$ ratio between any two terms.

Calculation:

Consider the sequence $1000,200,40,8,\mathrm{...}$

Put the values in the following formula,

  $t_n=t_1{r}^{n-1}$ ,

Where,

  $\begin{array}{l}{t}_1=1000\\ r=\frac{1}{5}\end{array}$

So, the result is, $t_n=1000{\left(\frac{1}{5}\right)}^{n-1}$

Now simplify the ${\left(\frac{1}{5}\right)}^{n-1}$

So, raise the $\left(\frac{1}{5}\right)$ to, each power separately and then multiplies.

  $t_n=1000{\left(\frac{1}{5}\right)}^{-1}{\left(\frac{1}{5}\right)}^n$

Then with the help of reciprocal of the number,

  ${\left(\frac{1}{5}\right)}^{-1}=5$

Now replacing ${\left(\frac{1}{5}\right)}^{-1}$ with $5$ , them simplify by multiplying $5$ by $1000$ to get final answer.

  $\begin{array}{l}{t}_n=1000\left(5\right){\left(\frac{1}{5}\right)}^n\\ t_n=5000{\left(\frac{1}{5}\right)}^n\end{array}$

So, the formula is $t_n=5000{\left(\frac{1}{5}\right)}^n$ .