Chapter 4.7, Problem 3P

(a)

To determine

The recursive formula and $9th$ term for the arithmetic sequence is to be calculated.

Answer to Problem 3P

The recursive formula for the given sequence is,

  $A\left(n\right)=A\left(1\right)+\left(n-1\right)6$

The $9th$ of the sequence is $51$ .

Explanation of Solution

The given arithmetic sequence is,

  $3,9,15,\mathrm{21......}$

The recursive formal is functions rule that relates the each term of the sequence.

The recursive formula for any arithmetic sequence can be written using the common difference of the terms.

The common difference for given arithmetic sequence = $6$

Therefore the recursive formula for the given sequence becomes,

  $A\left(n\right)=A\left(1\right)+\left(n-1\right)6$........(1)

Where,

• $A\left(n\right)$ represents the value of nth term.
• $A\left(1\right)$ represents the value of first term of the sequence.
• $n$ represents the number of term in the sequence.

The value of $9th$ term of the sequence is calculated as follows:

From the given arithmetic sequence,

  $A\left(1\right)=3$

  $d=6$

For $9th$ term,

  $n=9$

Plugin the values in equation (1)

  $\begin{array}{l}A\left(9\right)=3+\left(9-1\right)6\\ =3+48\\ =51\end{array}$

(b)

To determine

The recursive formula and $9th$ term for the arithmetic sequence is to be calculated.

Answer to Problem 3P

The recursive formula for the given sequence is,

  $A\left(n\right)=A\left(1\right)=23+\left(n-1\right)12$

The $9th$ of the sequence is $119$ .

Explanation of Solution

The given arithmetic sequence is,

  $23,35,47,\mathrm{59....}$

The recursive formal is functions rule that relates the each term of the sequence.

The recursive formula for any arithmetic sequence can be written using the common difference of the terms.

The common difference for given arithmetic sequence = $12$

Therefore the recursive formula for the given sequence becomes,

  $A\left(n\right)=A\left(1\right)+\left(n-1\right)12$........(1)

Where,

• $A\left(n\right)$ represents the value of nth term.
• $A\left(1\right)$ represents the value of first term of the sequence.
• $n$ represents the number of term in the sequence.

The value of $9th$ term of the sequence is calculated as follows:

From the given arithmetic sequence,

  $A\left(1\right)=23$

  $d=12$

For $9th$ term,

  $n=9$

Plugin the values in equation (1)

  $\begin{array}{l}A\left(9\right)=23+\left(9-1\right)12\\ =23+96\\ =119\end{array}$

(c)

To determine

The recursive formula and $9th$ term for the arithmetic sequence is to be calculated.

Answer to Problem 3P

The recursive formula for the given sequence is,

  $A\left(n\right)=A\left(1\right)=23+\left(n-1\right)0.5$

The $9th$ of the sequence is $11.3$ .

Explanation of Solution

The given arithmetic sequence is,

  $7.3,7.8,8.3,\mathrm{8.8....}$

The recursive formal is functions rule that relates the each term of the sequence.

The recursive formula for any arithmetic sequence can be written using the common difference of the terms.

The common difference for given arithmetic sequence = $0.5$

Therefore the recursive formula for the given sequence becomes,

  $A\left(n\right)=A\left(1\right)+\left(n-1\right)0.5$........(1)

Where,

• $A\left(n\right)$ represents the value of nth term.
• $A\left(1\right)$ represents the value of first term of the sequence.
• $n$ represents the number of term in the sequence.

The value of $9th$ term of the sequence is calculated as follows:

From the given arithmetic sequence,

  $A\left(1\right)=7.3$

  $d=0.5$

For $9th$ term,

  $n=9$

Plugin the values in equation (1)

  $\begin{array}{l}A\left(9\right)=7.3+\left(9-1\right)0.5\\ =7.3+4\\ =11.3\end{array}$

(d)

To determine

The recursive formula and $9th$ term for the arithmetic sequence is to be calculated.

Answer to Problem 3P

The recursive formula for the given sequence is,

  $A\left(n\right)=A\left(1\right)=23+\left(n-1\right)\times \left(-9\right)$

The $9th$ of the sequence is $25$ .

Explanation of Solution

The given arithmetic sequence is,

  $97,88,79,\mathrm{70....}$

The recursive formal is functions rule that relates the each term of the sequence.

The recursive formula for any arithmetic sequence can be written using the common difference of the terms.

The common difference for given arithmetic sequence = $-9$

Therefore the recursive formula for the given sequence becomes,

  $A\left(n\right)=A\left(1\right)+\left(n-1\right)\times \left(-9\right)$........(1)

Where,

• $A\left(n\right)$ represents the value of nth term.
• $A\left(1\right)$ represents the value of first term of the sequence.
• $n$ represents the number of term in the sequence.

The value of $9th$ term of the sequence is calculated as follows:

From the given arithmetic sequence,

  $A\left(1\right)=97$

  $d=-9$

For $9th$ term,

  $n=9$

Plugin the values in equation (1)

  $\begin{array}{l}A\left(9\right)=97+\left(9-1\right)\times \left(-9\right)\\ =97-72\\ =25\end{array}$

(e)

To determine

The recursive formula is to be analyzed.

Explanation of Solution

Sequence is an ordered list of numbers that form a pattern.

The sequence can be extended, once the pattern is identified.

To find a particular term of the sequence, it is not required to write the sequence up to that term. The particular term can be calculated using the recursive formula.

The sequence can be modeled with some function rule that function rule is known as the recursive rule.

Therefore the recursive formula is a useful way to find a value of an arithmetic sequence.