Chapter 8.5, Problem 18E

To determine

To write: a recursive rule for the given sequence.

Answer to Problem 18E

  $a_n=\left(a_{n-1}\right)\left(a_{n-2}\right)$ .

Explanation of Solution

Given:

The given sequence is $3,5,15,75,1125,\mathrm{......}$ .

Calculation:

According to the given sequence is $3,5,15,75,1125,\mathrm{......}$ .

Therefore, the terms of the given sequence are $a_1=3,a_2=5,a_3=15,\mathrm{.....}\text{ and so on}$ .

Now, from the given terms, it is found that

  $\begin{array}{l}{a}_1=3\{\text{given terms}}\\ a_n=\left(a_{n-1}\right)\left(a_{n-2}\right)\end{array}$

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

  $\begin{array}{l}{a}_n=\left(a_{n-1}\right)\left(a_{n-2}\right)\{\text{recursive rule}}\\ a_3=\left(a_{3-1}\right)\left(a_{3-2}\right)\{\text{put }n=3\text{}}\\ a_3=a_2\times a_1\\ a_3=5\times 3\{\text{given }{a}_1=3;a_2=5\text{}}\\ a_3=15\{\text{true}}\\ a_4=\left(a_{4-1}\right)\left(a_{4-2}\right)\{\text{put }n=4\text{}}\\ a_4=a_3\times a_2\\ a_4=15\times 5\{\text{given }{a}_3=15;a_2=5\text{}}\\ a_4=75\{\text{true}}\end{array}$

Hence, the recursive rule for the given sequence is $a_n=\left(a_{n-1}\right)\left(a_{n-2}\right)$ .