Chapter 12.3, Problem 86E

a.

To determine

To find if the given sequence is an arithmetic, geometric or neither, and find the next term of the sequence is arithmetic or geometric.

Answer to Problem 86E

Neither

Explanation of Solution

Given:

  $5,-3,5,-3,\cdot \cdot \cdot$

Calculation:

Observe that $-3-5=-8$ and $5-\left(-3\right)=8$ . So, the difference is not common. That means, it is not an arithmetic.

Also, $\frac{-3}{5}\ne \frac{5}{-3}$ . The ratio of the second to the first term and the ratio of the third term to the second term are not equal. So, the sequence not geometric as well.

b.

To determine

To find if the given sequence is an arithmetic, geometric or neither, and find the next term of the sequence is arithmetic or geometric.

Answer to Problem 86E

Arithmetic, 3.

Explanation of Solution

Given:

  $\frac{1}{3},1,\frac{5}{3},\frac{7}{3},\cdot \cdot \cdot$

Calculation:

Observe that $1-\frac{1}{3}=\frac{2}{3}$ and $\frac{5}{3}-1=\frac{2}{3}$ .

That is, each term differs by $\frac{2}{3}$ . So, the sequence is arithmetic.

Then, the next term is $\frac{7}{3}+\frac{2}{3}=\frac{9}{3}=3$ .

c.

To determine

To find if the given sequence is an arithmetic, geometric or neither, and find the next term of the sequence is arithmetic or geometric.

Answer to Problem 86E

Geometric, $9\sqrt{3}$ .

Explanation of Solution

Given:

  $\sqrt{3},3,3\sqrt{3},9,\cdot \cdot \cdot$

Calculation:

Observe that $\frac{3}{\sqrt{3}}=\sqrt{3}$ and $\frac{3\sqrt{3}}{3}=\sqrt{3}$ .

That is, the common ratio is $\sqrt{3}$ . So, the sequence is geometric.

Then, the next term is $9\cdot \sqrt{3}=9\sqrt{3}$ .

d.

To determine

To find if the given sequence is an arithmetic, geometric or neither, and find the next term of the sequence is arithmetic or geometric.

Answer to Problem 86E

Geometric, 1.

Explanation of Solution

Given:

  $1,-1,1,-1,\cdot \cdot \cdot$

Calculation:

Observe that $\frac{-1}{1}=-1$ and $\frac{1}{-1}=-1$ .

That is, the common ratio is $-1$ . So, the sequence is geometric.

Then, the next term is $-1\cdot \left(-1\right)=1$ .

e.

To determine

To find if the given sequence is an arithmetic, geometric or neither, and find the next term of the sequence is arithmetic or geometric.

Answer to Problem 86E

Neither

Explanation of Solution

Given:

  $2,-1,\frac{1}{2},2,\cdot \cdot \cdot$

Calculation:

Observe that $\frac{-1}{2}=-\frac{1}{2}$ and $\frac{\left(\frac{1}{2}\right)}{-1}=-\frac{1}{2}$ . But $\frac{2}{\left(\frac{1}{2}\right)}=1$ .

That means, the ratio is not common. So, the sequence is not geometric.

Also, $-1-2=-3$ and $\frac{1}{2}-\left(-1\right)=\frac{3}{2}$ . That is, the difference of terms is not equal. So, the sequence is neither arithmetic.

f.

To determine

To find if the given sequence is an arithmetic, geometric or neither, and find the next term of the sequence is arithmetic or geometric.

Answer to Problem 86E

Arithmetic, $x+3$ .

Explanation of Solution

Given:

  $x-1,x,x+1,x+2,\cdot \cdot \cdot$

Calculation:

Observe that $x-\left(x-1\right)=1$ and $\left(x+1\right)-x=1$ .

That is, each term differs by 1. So, the sequence is arithmetic.

Then, the next term is $\left(x+2\right)+1=x+3$ .

g.

To determine

To find if the given sequence is an arithmetic, geometric or neither, and find the next term of the sequence is arithmetic or geometric.

Answer to Problem 86E

Arithmetic, 3.

Explanation of Solution

Given:

  $-3,-\frac{3}{2},0,\frac{3}{2},\cdot \cdot \cdot$

Calculation:

Observe that $-\frac{3}{2}-\left(-3\right)=\frac{3}{2}$ , $0-\left(-\frac{3}{2}\right)=\frac{3}{2}$ and $\frac{3}{2}-0=\frac{3}{2}$ .

That is, each term differs by $\frac{3}{2}$ . So, the sequence is arithmetic.

Then, the next term is $\frac{3}{2}+\frac{3}{2}=3$ .

h.

To determine

To find if the given sequence is an arithmetic, geometric or neither, and find the next term of the sequence is arithmetic or geometric.

Answer to Problem 86E

Geometric, $\frac{1}{\sqrt[6]{5}}$ .

Explanation of Solution

Given:

  $\sqrt{5},\sqrt[3]{5},\sqrt[6]{5},1,\cdot \cdot \cdot$

Calculation:

Observe that $\frac{\sqrt[3]{5}}{\sqrt{5}}=5^{\frac{1}{3}-\frac{1}{2}}=5^{-\frac{1}{6}}=\frac{1}{\sqrt[6]{5}}$ and $\frac{\sqrt[6]{5}}{\sqrt[3]{5}}=5^{\frac{1}{6}-\frac{1}{3}}=5^{-\frac{1}{6}}=\frac{1}{\sqrt[6]{5}}$ .

That is, the common ratio is $\frac{1}{\sqrt[6]{5}}$ . So, the sequence is geometric.

Then, the next term is $1\cdot \frac{1}{\sqrt[6]{5}}=\frac{1}{\sqrt[6]{5}}$ .