Chapter 12.3, Problem 7E

a.

To determine

Calculate the first five terms of the sequence.

Answer to Problem 7E

The first five term of the sequence are $\frac{5}{2},-\frac{5}{4},\frac{5}{8},-\frac{5}{16},\frac{5}{16}$ .

Explanation of Solution

Given:

It is given in the question that the sequence is $a_n=\frac{5}{2}{(-\frac{1}{2})}^{n-1}$ .

Concept Used:

In this, use the concept of putting the value of counting numbers in the sequence to get the answer.

Calculation: Here, the sequence is $a_n=\frac{5}{2}{(-\frac{1}{2})}^{n-1}$ .

Put, $n=1,2,3,4,5$

  $\begin{array}{}$ a_n=\frac{5}{2}{(-\frac{1}{2})}^{n-1}nthterm$$ a_1=\frac{5}{2}{(-\frac{1}{2})}^{1-1}=\frac{5}{2}{(-\frac{1}{2})}^0=\frac{5}{2}(1)=\frac{5}{2};n=1$$ a_2=\frac{5}{2}{(-\frac{1}{2})}^{2-1}=\frac{5}{2}{(-\frac{1}{2})}^1=\frac{5}{2}(-\frac{1}{2})=\frac{-5}{4};n=2$$ a_3=\frac{5}{2}{(-\frac{1}{2})}^{3-1}=\frac{5}{2}{(-\frac{1}{2})}^2=\frac{5}{2}(\frac{1}{4})=\frac{5}{8};n=3$$ a_4=\frac{5}{2}{(-\frac{1}{2})}^{4-1}=\frac{5}{2}{(-\frac{1}{2})}^3=\frac{5}{2}(-\frac{1}{8})=-\frac{5}{16};n=4$$ a_5=\frac{5}{2}{(-\frac{1}{2})}^{5-1}=\frac{5}{2}{(-\frac{1}{2})}^4=\frac{5}{2}(\frac{1}{16})=\frac{5}{32};n=5$\end{array}$

Conclusion:

The first five term of the sequence are $\frac{5}{2},-\frac{5}{4},\frac{5}{8},-\frac{5}{16},\frac{5}{16}$ .

b.

To determine

Find the common ratio ( r ).

Answer to Problem 7E

The common ratio is $-\frac{1}{2}$ .

Explanation of Solution

Given:

It is given in the question that the terms are $\frac{5}{2},-\frac{5}{4},\frac{5}{8},-\frac{5}{16},\frac{5}{16}$ .

Concept Used:

In this, use the concept of geometric progression with formula $r=\frac{a_2}{a_1}$ .

Calculation: Here, the term are $\frac{5}{2},-\frac{5}{4},\frac{5}{8},-\frac{5}{16},\frac{5}{16}$ .

  $\begin{array}{l}r=\frac{a_2}{a_1}\\ r=-\frac{5}{4}÷\frac{5}{2}\\ r=-\frac{5}{4}\cdot \frac{2}{5}\\ r=-\frac{1}{2}\end{array}$

Conclusion:

The common ratio is $-\frac{1}{2}$ .

c.

To determine

Draw the graph of the terms.

Explanation of Solution

Given:

It is given in the question that the term are $\frac{5}{2},-\frac{5}{4},\frac{5}{8},-\frac{5}{16},\frac{5}{16}$ .

Graph:

  

Interpretation:

Here, put all the terms $\frac{5}{2},-\frac{5}{4},\frac{5}{8},-\frac{5}{16},\frac{5}{16}$ in the TI − eighty three calculator and it gives us the required graph as shown above with correct point.