Chapter 9.1, Problem 15E

(a).

To determine

To Find: The common ratio of a given geometric sequence.

Answer to Problem 15E

The common ratio of the sequence is $\frac{1}{2}$ .

Explanation of Solution

Given Information: The given geometric sequence is $8,4,2,1,\dots$

Concept Used: The common ratio of a geometric sequence is the ratio of any of its two consecutive terms.

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

  $r=\frac{t_2}{t_1}=\frac{4}{8}=\frac{1}{2}$

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

(b).

To determine

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

Answer to Problem 15E

The nineth term of the sequence is $\frac{1}{32}$

Explanation of Solution

Given Information: The first term of the sequence is $8$ and from the previous sub-part, the common ratio is $\frac{1}{2}$ .

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

Calculation:

  $\begin{array}{l}ninethterm=firstterm\times \stackrel{8times}{\overbrace{r\times r\times \dots \times r}}\\ =firstterm\times r^8\\ =8\times {\left(\frac{1}{2}\right)}^8\\ =\frac{1}{32}\end{array}$

(c).

To determine

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

Answer to Problem 15E

The series can be recursively defined as $a_1=8,a_n=a_{n-1}\times \frac{1}{2}$ for all $n\ge 2$ .

Explanation of Solution

Given Information: The sequence is $8,4,2,1,\dots$ and from the first sub-part, its common ratio is $\frac{1}{2}$ .

Concept Used: Each term in a geometric sequence (except the first term) is the product of its preceding term and the common ratio.

Calculation: The first term in the sequence is $8$ . So

  $a_1=8$

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

  $a_n=precedingterm\times commonratio=a_{n-1}\times \frac{1}{2}$

Therefore, the sequence can be defined recursively as $a_1=8,a_n=a_{n-1}\times \frac{1}{2}$ for all $n\ge 2$ .

(d).

To determine

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

Answer to Problem 15E

The $n^{th}$ term of the given sequence is $8\cdot {\left(\frac{1}{2}\right)}^{n-1}$

Explanation of Solution

Given Information: The first term of the given sequence is $8$ and from the first sub-part, the common ratio is $\frac{1}{2}$

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

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

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

  $=a_1{r}^{n-1}$

  $=8\cdot {\left(\frac{1}{2}\right)}^{n-1}$

Therefore, the $n^{th}$ term of the given sequence is $=8\times {\left(\frac{1}{2}\right)}^{n-1}$ .