Chapter 4, Problem 27CR

To determine

To find:A recursive and an explicit formula for the sequence $18,11,4,-3,\mathrm{...}$

Answer to Problem 27CR

The recursive formula is $A(n)=A(n-1)-7$ , where $A(1)=18$ .

The explicit formula is $A(n)=18-(n-1)7$ .

Explanation of Solution

Given information: $18,11,4,-3,\mathrm{...}$

Calculation:

Recursive formula:

The common difference of the arithmetic sequence is $d=11-18=-7$

  $\begin{array}{l}A(1)=18\\ A(2)=A(1)-7=18-7=11\\ A(3)=A(2)-7=11-7=4\\ A(4)=A(3)-7=4-7=-3\\ A(n)=A(n-1)-7\end{array}$

The recursive formula for the arithmetic sequence is $A(n)=A(n-1)-7$ , where $A(1)=18$ .

Explicit Formula:

The general form of an explicit formula is,

  $A(n)=A(1)+(n-1)d$

Substituting $A(1)=18$ and $d=-7$ ,

  $\begin{array}{l}A(n)=A(1)+(n-1)d\\ \Rightarrow A(n)=18+(n-1)(-7)\\ \Rightarrow A(n)=18-(n-1)7\end{array}$

The explicit formula for the arithmetic sequence is $A(n)=18-(n-1)7$ .