Chapter 4.7, Problem 5P

a.

To determine

To find : an explicit formula that represents the same sequence..

Answer to Problem 5P

  $A\left(n\right)=21+\left(n-1\right)\left(2\right)$

Explanation of Solution

Given information: The recursive formula for the sequence is $A\left(n\right)=A\left(n-1\right)+2;A\left(1\right)=21$

Calculation:

The first term of given sequence is $A\left(1\right)=21$

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

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

Therefore, the common ratio is $d=2$ .

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

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

Thus, explicit formula for given sequence is

  $A\left(n\right)=21+\left(n-1\right)\left(2\right)$

b

To determine

To find : an explicit formula that represents the same sequence..

Answer to Problem 5P

  $A\left(n\right)=2+\left(n-1\right)\left(7\right)$

Explanation of Solution

Given information: The recursive formula for the sequence is $A\left(n\right)=A\left(n-1\right)+7;A\left(1\right)=2$

Calculation:

The first term of given sequence is $A\left(1\right)=2$

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

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

Therefore, the common ratio is $d=7$ .

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

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

Thus, explicit formula for given sequence is

  $A\left(n\right)=2+\left(n-1\right)\left(7\right)$