Chapter 4.7, Problem 6P

a.

To determine

To find : a recursive formula that represents the same sequence..

Answer to Problem 6P

  $A\left(n\right)=A\left(n-1\right)+10;A\left(1\right)=76$

Explanation of Solution

Given information: The explicit formula for the sequence is $A\left(n\right)=76+\left(n-1\right)\left(10\right)$

Calculation:

We know that 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, the first term of given sequence is $A\left(1\right)=76$ and common ratio is $d=10$ .

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

  $A\left(n\right)=A\left(n-1\right)+10;A\left(1\right)=76$

b

To determine

To find : a recursive formula that represents the same sequence..

Answer to Problem 6P

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

Explanation of Solution

Given information: The explicit formula for the sequence is $A\left(n\right)=1+\left(n-1\right)\left(3\right)$

Calculation:

We know that 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, the first term of given sequence is $A\left(1\right)=1$ and common ratio is $d=3$ .

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

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