Chapter 4.7, Problem 59PPE

To determine

whether the sequence $0.2,-0.6,-1.4,-2.2$ is arithmetic and if it is ,then find recursive formula and explicit formula for it.

Answer to Problem 59PPE

The sequence is an arithmetic sequence.

Explicit formula: $A\left(n\right)=0.2+\left(n-1\right)\left(-0.8\right)$

Recursive formula: $A\left(n\right)=A\left(n-1\right)-0.8;A\left(1\right)=0.2$

Explanation of Solution

Given sequence: $0.2,-0.6,-1.4,-2.2$

Calculation:

The difference of the terms with the previous term is

  $\begin{array}{l}-0.6-0.2=-0.8\\ -1.4-\left(-0.6\right)=-0.8\\ -2.2-\left(-1.4\right)=-0.8\end{array}$

Since each term of given sequence after the first term is obtained by adding the previous one by a fixed constant, that is $-0.8$ , therefore the sequence is an arithmetic sequence and the common difference is d = $-0.8$ .

The first term of the sequence is $A\left(1\right)=0.2$

The explicit formula for the arithmetic sequence with first term $A\left(1\right)$ and common ratio d is

  $A\left(n\right)=A\left(1\right)+\left(n-1\right)\left(d\right)$

Therefore, explicit formula for given sequence is

  $A\left(n\right)=0.2+\left(n-1\right)\left(-0.8\right)$

Now, the recursive formula for the arithmetic sequence with first term $A\left(1\right)$ and common ratio d is

  $A\left(n\right)=A\left(n-1\right)+d$

Thus, recursive formula for given sequence is

  $A\left(n\right)=A\left(n-1\right)-0.8;A\left(1\right)=0.2$