Chapter 11.1, Problem 9WE

To determine

To calculate:

The following sequence is arithmetic, geometric or neither and the missing terms:

  $4^{1/2},4^{3/2},4^{5/2},4^{7/2},?,?$

Answer to Problem 9WE

The sequence is geometric sequence and the missing terms are $512$ and $2048$ .

Explanation of Solution

Given information:

  $4^{1/2},4^{3/2},4^{5/2},4^{7/2},?,?$

Calculation:

Check the common difference first to categorize the given sequence into arithmetic, geometric or neither:

  $\begin{array}{l}{a}_1=4^{1/2}\\ a_2=4^{3/2}\\ a_3=4^{5/2}\\ a_4=4^{7/2}\end{array}$

Now, the common difference is calculated as:

  $\begin{array}{l}r=a_2-a_1=4^{3/2}-4^{1/2}=8-2=6\\ r=a_3-a_2=4^{5/2}-4^{3/2}=32-8=24\\ r=a_4-a_3=4^{7/2}-4^{5/2}=128-32=96\end{array}$

Therefore, the given sequence is not arithmetic sequence as the common ratio is not same.

Now, check for geometric sequence:

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{4^{3/2}}{4^{1/2}}=\frac{8}{2}=4\\ r=\frac{a_3}{a_2}=\frac{4^{5/2}}{4^{3/2}}=\frac{32}{8}=4\\ r=\frac{a_4}{a_3}=\frac{4^{7/2}}{4^{5/2}}=\frac{128}{32}=4\end{array}$

Therefore, the given sequence is geometric sequence. Now, the missing terms are calculated as:

  $\begin{array}{l}{a}_5=a{r}^{\left(5-1\right)}=\left(4^{1/2}\right){\left(4\right)}^{\left(4\right)}=512\\ a_6=a{r}^{\left(6-1\right)}=\left(4^{1/2}\right){\left(4\right)}^{\left(5\right)}=2048\end{array}$