Chapter 9, Problem 5PS

(a)

To determine

To find:If the resulting sequence from adding a constant C to each term of an arithmetic sequence is arithmetic or not.

Answer to Problem 5PS

The resulting sequence is arithmetic.

Explanation of Solution

Given:

An arithmetic sequence and a constant C .

Calculation:

Let the first term of the given sequence be a and the common difference be d . Then the $n^{th}$ term of the sequence is given by,

  $\begin{array}{c}{a}_n=a+\left(n-1\right)d\\ =\left(a+nd\right)-d\\ =a_{n+1}-d\\ a_{n+1}-a_n=d\end{array}$  ...(1)

Let the new sequence be $\left(s_n\right)=\left(a_n+C\right)$ . Then the common difference is calculated by subtracting the ${\left(n+1\right)}^{th}$ term from the $n^{th}$ term, as shown below:

  $\begin{array}{c}{s}_{n+1}-s_n=\left(a_{n+1}+C\right)-\left(a_n+C\right)\\ =a_{n+1}+C-a_n-C\\ =a_{n+1}-a_n\\ =d\left(\text{from }\left(1\right)\right)\end{array}$

Since the common difference of the new series is also d , the series $\left(s_n\right)=\left(a_n+C\right)$ is also arithmetic.

(b)

To determine

To find:If the resulting sequence from multiplying a nonzero constant C to each term of an arithmetic sequence is arithmetic or not.

Answer to Problem 5PS

The resulting sequence is arithmetic.

Explanation of Solution

Given:

An arithmetic sequence and a constant C .

Calculation:

Let the first term of the given sequence be a and the common difference be d . Then the $n^{th}$ term of the sequence is given by,

  $\begin{array}{c}{a}_n=a+\left(n-1\right)d\\ =\left(a+nd\right)-d\\ =a_{n+1}-d\\ a_{n+1}-a_n=d\end{array}$  ...(1)

Let the new sequence be $\left(s_n\right)=\left(C{a}_n\right)$ . Then the common difference is calculated by subtracting the ${\left(n+1\right)}^{th}$ term from the $n^{th}$ term, as shown below:

  $\begin{array}{c}{s}_{n+1}-s_n=\left(C{a}_{n+1}\right)-\left(C{a}_n\right)\\ =C\left(a_{n+1}-a_n\right)\\ =Cd\left(\text{from }\left(1\right)\right)\end{array}$

Since the common difference of the new series isCd which is a constant nonzero number, the series $\left(s_n\right)=\left(C{a}_n\right)$ is also arithmetic.

(c)

To determine

To find:If the resulting sequence from squaring each term of an arithmetic sequence is arithmetic or not.

Answer to Problem 5PS

The resulting sequence is not arithmetic.

Explanation of Solution

Given:

An arithmetic sequence.

Calculation:

Let the first term of the given sequence be a and the common difference be d . Then the $n^{th}$ term of the sequence is given by,

  $\begin{array}{c}{a}_n=a+\left(n-1\right)d\\ =\left(a+nd\right)-d\\ =a_{n+1}-d\\ a_{n+1}-a_n=d\end{array}$  ...(1)

Let the new sequence be $\left(s_n\right)=\left(a_n{}^2\right)$ . Then the common difference is calculated by subtracting the ${\left(n+1\right)}^{th}$ term from the $n^{th}$ term, as shown below:

  $\begin{array}{c}{s}_{n+1}-s_n=a_{n+1}{}^2-a_n{}^2\\ ={\left(a_n+d\right)}^2-a_n{}^2\left(\text{from }\left(1\right)\right)\\ =a_n{}^2+d^2+2d{a}_n-a_n{}^2\\ =d^2+2d{a}_n\end{array}$

Since the common difference of the new series is not a constant term, the series $\left(s_n\right)=\left(a_n{}^2\right)$ is not arithmetic.