Chapter 4.7, Problem 55PPE

To determine

whether the sequence $-3,-7,-11,-15,\mathrm{......}$ is arithmetic and if it is ,then find recursive formula and explicit formula for it.

Answer to Problem 55PPE

The sequence is an arithmetic sequence.

Explicit formula: $A\left(n\right)=0-3+\left(n-1\right)\left(-4\right)$

Recursive formula: $A\left(n\right)=A\left(n-1\right)-4;A\left(1\right)=-3$

Explanation of Solution

Given sequence: $-3,-7,-11,-15,\mathrm{......}$

Calculation:

The difference of the terms with the previous term is

  $\begin{array}{l}-7-\left(-3\right)=-4\\ -11-\left(-7\right)=-4\\ -15-\left(-11\right)=-4\end{array}$

Since each term of given sequence after the first term is obtained by adding the previous one by a fixed constant, that is $-4$ , therefore the sequence is an arithmetic sequence and the common difference is d = $-4$ .

The first term of the sequence is $A\left(1\right)=-3$

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

  $A\left(n\right)=0-3+\left(n-1\right)\left(-4\right)$

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

  $A\left(n\right)=A\left(n-1\right)-4;A\left(1\right)=-3$