Chapter 12.3, Problem 37AYU

To determine

To find: The $n\text{th}$ term of the geometric sequence.

Answer to Problem 37AYU

The $n\text{th}$ term is $a_n=\left(\frac{7}{15}\right){\left(15\right)}^{n-1}$ .

Explanation of Solution

Given information:

  $a_2=7$ and $a_4=1575$ .

Calculation:

The first term of a sequence is denoted as $a_1$ and the second term of the sequence is denoted by $a_2$ .

The formula for nth term of geometric sequence is given by,

  $a_n=a_1{r}^{n-1}$

Calculate $a_2$ .

  $\begin{array}{l}{a}_2=a_1{r}^1\\ 7=a_{1}r\\ a_1=\frac{7}{r}\end{array}$ …… (1)

Calculate $a_4$ .

  $a_4=a_1{r}^3$

Substitute $\frac{7}{r}$ for $a_1$ in above equation.

  $\begin{array}{l}1575=\frac{7}{r}{r}^3\\ r^2=\frac{1575}{7}\\ r=15\end{array}$

Substitute $15$ for $r$ in equation (1).

  $a_1=\frac{7}{15}$

Calculate the $n\text{th}$ term.

  $a_n=\left(\frac{7}{15}\right){\left(15\right)}^{n-1}$

Therefore, the $n\text{th}$ term of the sequence is $a_n=\left(\frac{7}{15}\right){\left(15\right)}^{n-1}$ .