Chapter 8.5, Problem 21E

To determine

To write: a recursive rule for the given sequence.

Answer to Problem 21E

  $a_n=n{a}_{n-1}$ .

Explanation of Solution

Given:

The given sequence is $6,12,32,144,720,\mathrm{......}$ .

Calculation:

According to the given sequence is $6,12,32,144,720,\mathrm{......}$ .

Therefore, the terms of the given sequence are $a_1=6,a_2=12,a_3=36,\mathrm{.....}\text{ and so on}$ .

Now, from the given terms, it is found that

  $\begin{array}{l}{a}_1=6\{\text{given terms}}\\ a_n=n{a}_{n-1}\end{array}$

Now, check this regressive rule for 2^nd and 3^rd term and so on.

  $\begin{array}{l}{a}_n=n{a}_{n-1}\{\text{recursive rule}}\\ a_2=2a_{2-1}\{\text{put }n=2\text{}}\\ a_2=2a_1\\ a_2=2\left(6\right)\{\text{given }{a}_1=6\text{}}\\ a_2=12\{\text{true}}\\ a_n=n{a}_{n-1}\{\text{put }n=3\text{}}\\ a_3=3a_{3-1}\\ a_3=3a_2\{\text{given }{a}_3=12\}\\ a_3=3\left(12\right)\\ a_3=36\{\text{true}}\end{array}$

Hence, the recursive rule for the given sequence is $a_n=n{a}_{n-1}$ .