Chapter 4, Problem 25CR

To determine

To find:A recursive and an explicit formula for the sequence $-2,-5,-8,-11,\mathrm{...}$

Answer to Problem 25CR

The recursive formula is $A(n)=A(n-1)-3$ , where $A(1)=-2$ .

The explicit formula is $A(n)=-2-(n-1)3$ .

Explanation of Solution

Given information: $-2,-5,-8,-11,\mathrm{...}$

Calculation:

Recursive formula:

The common difference of the arithmetic sequence $=-5-(-2)=-5+2=-3$

  $\begin{array}{l}A(1)=-2\\ A(2)=A(1)-3=-2-3=-5\\ A(3)=A(2)-3=-5-3=-8\\ A(4)=A(3)-3=-8-3=-11\\ A(n)=A(n-1)-3\end{array}$

The recursive formula for the arithmetic sequence is $A(n)=A(n-1)-3$ , where $A(1)=-2$ .

Explicit Formula:

The general form of an explicit formula is,

  $A(n)=A(1)+(n-1)d$

Substituting $A(1)=-2$ and $d=-3$ ,

  $\begin{array}{l}A(n)=A(1)+(n-1)d\\ \Rightarrow A(n)=-2+(n-1)(-3)\\ \Rightarrow A(n)=-2-(n-1)3\end{array}$

The explicit formula for the arithmetic sequence is $A(n)=-2-(n-1)3$ .