Chapter 9.1, Problem 21E

To determine

To Find:The parameters of a geometric sequence, two of whose terms are given.

Answer to Problem 21E

The first term is $3.010$ , the common ratio is $10$ and the recursive rule for the $n^{th}$ term is $a_1=3.010,a_n=a_{n-1}\times 10$ for all $n\ge 2$ .

Explanation of Solution

Given Information: The $4^{th}$ term and the $7^{th}$ term of the geometric sequence are $3010$ and $3,010,000$ respectively.

Concept Used:

• The $n^{th}$ term of a geometric series with first term $a_1$ and common ratio $r$ is given by
$a_n=a_1{r}^{n-1}$
• Each term in a geometric sequence (except the first term) is the product of its preceding term and the common ratio.

Calculation:The fourth term is $a_4=3010$ . So

  $a_5=a_1{r}^{4-1}=3010$

  $\Rightarrow a_1{r}^3=3010$ (1)

The seventh term is $3,010,000$ . So

  $a_7=a_1{r}^{7-1}=3,010,000$

  $\Rightarrow a_1{r}^6=3,010,000$ (2)

Now, divideequation (1) by equation (2)to write

  $\frac{a_1{r}^6}{a_1{r}^3}=\frac{3,010,000}{3010}$

  $\Rightarrow r^3=1000$

  $\Rightarrow r=10$

Put the calculated value $r=10$ in equation (1) to write

  $a_1\times {10}^3=3010$

  $\Rightarrow a_1=3.010$

Therefore, the first term is $3.010$ , the common ratio is $10$ and the recursive rule for the $n^{th}$ term is $a_1=3.010,a_n=a_{n-1}\times 10$ for all $n\ge 2$ .