Chapter 11.1, Problem 3WE

To determine

To calculate:

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

  $1,5,25,125?,?$

Answer to Problem 3WE

The sequence is geometric sequence and the missing terms are $625$ and $3125$ .

Explanation of Solution

Given information:

  $1,5,25,125?,?$

Calculation:

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

  $\begin{array}{l}{a}_1=1\\ a_2=5\\ a_3=25\\ a_4=125\end{array}$

Now, the common difference is calculated as:

  $\begin{array}{l}r=a_2-a_1=5-1=4\\ r=a_3-a_2=25-5=20\\ r=a_4-a_3=125-25=100\end{array}$

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

Now, check for geometric sequence:

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{5}{1}=5\\ r=\frac{a_3}{a_2}=\frac{25}{5}=5\\ r=\frac{a_4}{a_3}=\frac{125}{25}=5\end{array}$

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

  $\begin{array}{l}{a}_5=a{r}^{\left(5-1\right)}=\left(1\right){\left(5\right)}^{\left(4\right)}=625\\ a_6=a{r}^{\left(6-1\right)}=\left(1\right){\left(5\right)}^{\left(5\right)}=3125\end{array}$