Chapter 8.5, Problem 17E

To determine

To write:a recursive rulefor the given sequence.

Answer to Problem 17E

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

Explanation of Solution

Given:

The givensequence is $2,5,10,50,500,\mathrm{......}$ .

Calculation:

According to the given sequence is $2,5,10,50,500,\mathrm{......}$ .

Therefore, the terms of the given sequenceare $a_1=2,a_2=5,a_3=10,\mathrm{.....}\text{ and so on}$ .

Now, from the given terms, it is found that

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

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

  $\begin{array}{l}{a}_n=\left(a_{n-1}\right)\left(a_{n-2}\right)\{\text{recursive rule}}\\ a_3=\left(a_{3-1}\right)\left(a_{3-2}\right)\{\text{put }n=3\text{}}\\ a_3=a_2\times a_1\\ a_3=5\times 2\{\text{given }{a}_1=2;a_2=5\text{}}\\ a_3=10\{\text{true}}\\ a_4=\left(a_{4-1}\right)\left(a_{4-2}\right)\{\text{put }n=4\text{}}\\ a_4=a_3\times a_2\\ a_4=10\times 5\{\text{given }{a}_3=10;a_2=5\text{}}\\ a_4=50\{\text{true}}\end{array}$

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