Chapter 4.7, Problem 54PPE

To determine

whether the sequence $0.3,0.9,1.5,2.1,\mathrm{......}$ is arithmetic and if it is ,then find recursive formula and explicit formula for it.

Answer to Problem 54PPE

The sequence is an arithmetic sequence.

Explicit formula: $A\left(n\right)=0.3+\left(n-1\right)\left(0.6\right)$

Recursive formula: $A\left(n\right)=A\left(n-1\right)+0.6;A\left(1\right)=0.3$

Explanation of Solution

Given sequence: $0.3,0.9,1.5,2.1,\mathrm{......}$

Calculation:

The difference of the terms with the previous term is

  $\begin{array}{l}0.9-0.3=0.6\\ 1.5-0.9=0.6\\ 2.1-1.5=0.6\end{array}$

Since each term of given sequence after the first term is obtained by adding the previous one by a fixed constant, that is $0.6$ , therefore the sequence is an arithmetic sequence and the common difference is d = $0.6$ .

The first term of the sequence is $A\left(1\right)=0.3$

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

  $A\left(n\right)=0.3+\left(n-1\right)\left(0.6\right)$

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

  $A\left(n\right)=A\left(n-1\right)+0.6;A\left(1\right)=0.3$