Chapter 6.7, Problem 22E

To determine

To write: A recursive rule for the given sequence

Answer to Problem 22E

  $a_n=a_{n-1}+\frac{1}{4},a_1=\frac{9}{2}$

Explanation of Solution

Given: The length of the deer antler grows as

n	1	2	3	4
$a_n$	$4\frac{1}{2}=\frac{9}{2}$	$4\frac{3}{4}=\frac{19}{4}$	5	$5\frac{1}{4}=\frac{21}{4}$

where n is in days.

The value of $a_n$ for $n=1$ is $\frac{9}{2}$ . Hence we must have $a_1=\frac{9}{2}$

It can be observed that the given values of $a_n$ forms an arithmetic progression with a common difference of $\frac{1}{4}$ .

A recursive relation for an arithmetic progression with common difference d is given by $a_n=a_{n-1}+d$ .

Hence, for the given problem the recursive relation must be given by:

  $a_n=a_{n-1}+\frac{1}{4},a_1=\frac{9}{2}$ .