Chapter 8.5, Problem 13E

To determine

To write: a recursive rule for the given sequence.

Answer to Problem 13E

  $a_n=4a_{n-1}$ .

Explanation of Solution

Given:

The given sequence is $3,12,18,192,768,\mathrm{......}$ .

Calculation:

According to the given sequence is $3,12,18,192,768,\mathrm{......}$ .

Therefore, the terms of the given sequence are $a_1=3,a_2=12,a_3=48,\mathrm{.....}\text{ and so on}$ .

Now, from the given terms, it is found that

  $\begin{array}{l}{a}_1=3\{\text{given terms}}\\ a_n=4a_{n-1}\end{array}$

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

  $\begin{array}{l}{a}_n=4a_{n-1}\{\text{recursive rule}}\\ a_2=4a_{2-1}\{\text{put }n=2\text{}}\\ a_2=4a_1\\ a_2=4\left(3\right)\{\text{given }{a}_1=3\text{}}\\ a_2=12\{\text{true}}\\ a_3=4a_{3-1}\{\text{put }n=3\text{}}\\ a_3=4a_2\\ a_3=4\left(12\right)\{\text{given }{a}_2=12\text{}}\\ a_3=48\{\text{true}}\end{array}$

Hence, the recursive rule for the given sequence is $a_n=4a_{n-1}$ .