Chapter 7.10, Problem 36PFA

a.

To determine

To identify: The type of sequence.

Answer to Problem 36PFA

recursive.

Explanation of Solution

Given: MULTI-STEP The table shows several terms of a sequence.

  

Calculation:

Since $a_2-a_1=25-(-12)=37and{a}_3-a_2=-49-25=-74\ne 37$ , so the sequence does not have common difference, so it is not arithmetic.

Since $\frac{a_2}{a_1}=\frac{25}{-12}=-\frac{25}{12}and\frac{a_3}{a_2}=\frac{-49}{25}\ne -\frac{25}{12}$ , so there is no common ratio .so it is not geometric.

Note however that if we multiply each term by -2 and then add 1,

we get the next term since

  $\begin{array}{c}\Rightarrow -2\left(-12\right)+1\\ =24+1=25,\\ \Rightarrow -2\left(25\right)+1\\ =-50+1\\ =-49,\text{ and}\\ \Rightarrow -2\left(-49\right)+1\\ =98+1\\ =99.\end{array}$

Since each term then depends on the previous term, it can be written as a recursive sequence.

b.

To determine

To find: The next term in the sequence.

Answer to Problem 36PFA

Next term of pattern is -197.

Explanation of Solution

Given: The table shows several terms of a sequence.

  

Calculation:

From part (a), the pattern is multiply by -2 and then add 1, we get the next term is ,

- $\begin{array}{c}2\left(99\right)+1\\ =-198+1\\ =-197\end{array}$

Next term of pattern is -197.

c.

To determine

To find: The formula which can describe the sequence.

Answer to Problem 36PFA

  $a_1=-12,a_n=-2a_{n-1}+1,n\ge 2$ .

Explanation of Solution

Given: The table shows several terms of a sequence.

  

Calculation:

we multiply each term by -2 and then add 1, we get the next term since

  $\begin{array}{c}\Rightarrow -2\left(-12\right)+1\\ =24+1=25,\\ \Rightarrow -2\left(25\right)+1\\ =-50+1\\ =-49,\text{ and}\\ \Rightarrow -2\left(-49\right)+1\\ =98+1\\ =99.\end{array}$

Since each term then depends on the previous term, it can be written as a recursive sequence.

  $a_1=-12,a_n=-2a_{n-1}+1,n\ge 2$ .

The correct answer is then choice B.

Conclusion:

d.

To determine

To explain: Whether an explicit formula be written as a linear equation for the sequence or not.

Answer to Problem 36PFA

No.

Explanation of Solution

Given: The table shows several terms of a sequence.

  

Calculation:

For an explicit formula of a sequence to be written as a linear equation, the sequence must be arithmetic since both athematic sequence and linear function need a constant rate of change (a constant difference), we know this sequence is not arithmetic so it cannot be written as a linear equation.