Chapter 9.1, Problem 13E

(a).

To determine

To Find: The common difference of a given arithmetic sequence.

Answer to Problem 13E

The common difference of the sequence is 3

  $\frac{1}{2}$ .

Explanation of Solution

Given Information: The given arithmetic sequence is $1,\frac{3}{2},2,\frac{5}{2},\dots$

Concept Used: The common difference of an arithmetic sequence is the difference between its any two consecutive terms.

Calculation: The difference between the second and first terms (or any other consecutive pair) is

  $d=t_2-t_1=\frac{3}{2}-1=\frac{1}{2}$

Therefore, the common difference of the sequence is $\frac{1}{2}$ .

(b).

To determine

To Find: The $9^{th}$ term of a given arithmetic sequence.

Answer to Problem 13E

The nineth term of the sequence is $5$

Explanation of Solution

Given Information: The first term of the sequence is $1$ and from the previous sub-part, the common difference is $\frac{1}{2}$ .

Concept Used: Add the common difference $8$ times consecutively to the $1^{st}$ term to get the $9^{th}$ term.

Calculation:

  $NinethTerm=FirstTerm+\stackrel{8times}{\overbrace{d+d+\dots +d}}$

  $=1+8\times \frac{1}{2}$

  $=5$

(c).

To determine

To Find: The recursive rule to find the $n^{th}$ term of a given arithmetic sequence.

Answer to Problem 13E

The sequence can be recursively defined as $a_1=1,a_n=a_{n-1}+\frac{1}{2}$ for all $n\ge 2$ .

Explanation of Solution

Given Information: The sequence is $1,\frac{3}{2},2,\frac{5}{2},\dots$ and from the first sub-part, its common difference is $\frac{1}{2}$

Concept Used: Each term in an arithmetic sequence (except the first term) is the sum of its preceding term and the common difference.

Calculation: The first term in the sequence is $1$ . So

  $a_1=1$

The common difference is $\frac{1}{2}$ . So, the $n^{th}$ term is given by

  $a_n=precedingterm+commondifference=a_{n-1}+\frac{1}{2}$

Therefore, the sequence can be defined recursively as $a_1=1,a_n=a_{n-1}+\frac{1}{2}$ for all $n\ge 2$ .

(d).

To determine

To Find: The explicit rule to calculate the $n^{th}$ term of a given arithmetic sequence.

Answer to Problem 13E

The $n^{th}$ term of the given sequence is $1+\frac{(n-1)}{2}$ .

Explanation of Solution

Given Information: The first term of the given sequence is $1$ and from the first sub-part, the common difference is $\frac{1}{2}$

Concept Used: The $n^{th}$ term of an arithmetic sequence can be calculated by adding the common difference $n-1$ times to the $1^{st}$ term.

Calculation: If the $n^{th}$ term is $a_n$ , $1^{st}$ term is $a_1$ and the common difference is $d$ , then

  $a_n=a_1+\stackrel{n-1times}{\overbrace{d+d+\dots +d}}$

  $=a_1+\left(n-1\right)d$

  $=1+\frac{(n-1)}{2}$

Therefore, the $n^{th}$ term of the given sequence is $1+\frac{(n-1)}{2}$ .