Chapter 4.7, Problem 1MP

(a)

To determine

To fill the blank space:

A(n) _____ is an ordered list of numbers that often form a pattern.

Answer to Problem 1MP

Sequence.

Explanation of Solution

Definition of sequence:

A sequence is an ordered list of numbers that often form a pattern. Each number in the list is called a term of a sequence.

(b)

To determine

To fill the blank space:

The function modeling Keiko’s blog is a(n) ________ formula

Answer to Problem 1MP

Explicit.

Explanation of Solution

Given information: The function model for the number of subscribes K, to the Keiko’s blog is

  $K=m^2+10$wherem denotes the number of months since she started the blog.

Formula Used:

Definition of Explicit Formula:

An explicit formula designates the $n^{th}$ term of the sequence as an expression in n

(wheren = the term’s location) .It defines the sequence as a formula in terms of n.

Calculation:

The table for the number of subscribers to Keiko’s blog for first few month is:

Number ofMonths	Number of Subscribes
0	10
1	11
2	14
3	19
4	26

The function model for the number of subscribes to Keiko’s blog can be written as

10, 11, 14, 19, 26…

∴The function modeling Keiko’s blog is an explicit formula since, the sequence is expressed in terms of a variable m(where m denotes number of months)

(c)

To determine

To fill the blank space:

The common difference in the function modeling Jayden’s blog is _________

Answer to Problem 1MP

8.

Explanation of Solution

Given Information:

The table for the number of subscribers to Jayden’s blog for first few months is:

Number ofMonths	Number of Subscribes
0	48
1	56
2	64
3	72
4	80

The sequential model for the number of subscribes to Jayden’s blog can be written as

48, 56, 64, 72, 80……

Common difference can be calculated by subtracting the $n^{th}$ term from ${\left(n-1\right)}^{th}$ term

  $\underset{8}{\underbrace{4856}}\underset{8}{\underbrace{64}}\underset{8}{\underbrace{72}}\underset{8}{\underbrace{80}}$ ....

∴the common difference in sequence of Jayden’s blog is 8

(d)

To determine

To fill the blank space:

Jayden’s blog will first have at least 100 subscribers in ______ months

Answer to Problem 1MP

8.

Explanation of Solution

Given Information:

The sequential model for the number of subscribes to Jayden’s blog is

48, 56, 64, 72, 80……

First term, $A\left(1\right)=48$

Common difference, $d=8$ (From part (c)) $$

  $n^{th}$ term, $A(n)=100$

Formula Used:

Explicit formula for an Arithmetic sequence:

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

Where, $A(n)$ denotes the $n^{th}$ term

  $A\left(1\right)$ denotes the first term.

d denotes the common difference

Calculation:

Substituting the values of $A(n)$ , d and $A\left(1\right)$ in $A(n)=A\left(1\right)+\left(n-1\right)d$

  $100=48+\left(n-1\right)8$

  $\left(n-1\right)8=100-48$

  $=52$

  $\left(n-1\right)=\frac{52}{8}$

  $n=\frac{52}{8}+1$

  $n=\frac{60}{8}$

  $n=7\frac{1}{2}\approx 8$

∴Jayden’s blog will first have at least 100 subscribers in 8 months

(e)

To determine

To fill the blank space:

Kelko’s blog will first have at least 100 subscribers in ______ months

Answer to Problem 1MP

10.

Explanation of Solution

Given Information: The function model for the number of subscribes K, to the Keiko’s blog is

  $K=m^2+10$wherem denotes the number of months since she started the blog.

Also Number of subscribers, $K=100$

Calculation:

  $K=m^2+10$

Substituting the value ofKinabove equation

  $100=m^2+10$

  $\begin{array}{c}{m}^2=100-10\\ =90\end{array}$

  $m=\sqrt{90}$

  $m=9.486\approx 10$

∴Kelko’s blog will first have at least 100 subscribers in 10 months.

(f)

To determine

To fill the blank space:

A function rule that relates each term of a sequence after the first term to the ones before it is called a(n) _____ formula

Answer to Problem 1MP

Recursive.

Explanation of Solution

Given Information

Recursive Formula:

A recursive formula is a function rule that defines each term of a sequence using preceding terms. Recursive formulas must always state the initial term, or terms, of the sequence.