Chapter 9.5, Problem 40PPSE

Solution

a.

To write the first four sums of the series.

The first four sums are $S_1=20,S_2=56,S_3=72.2$ and $S_4=72.746$ .

Given information:

The given graph models the sum of the first n terms of the geometric series with $a_1=20$ and $r=0.9$ .

Calculation:

The geometric series with $a_1=20$ and $r=0.9$ can be written as $20+20\left(\frac{9}{10}\right)+20\left(\frac{81}{100}\right)+20\left(\frac{243}{1000}\right)+\mathrm{...}$ .

Now, find the first four sums of the series.

  $\begin{array}{l}{S}_1=20\\ S_2=20+20\left(\frac{9}{10}\right)=20+20\times 1.8=20+36=56\\ S_3=20+20\left(\frac{9}{10}\right)+20\left(\frac{81}{100}\right)=56+20\times 0.81=56+16.2=72.2\\ S_4=20+20\left(\frac{9}{10}\right)+20\left(\frac{81}{100}\right)+20\left(\frac{243}{1000}\right)=72.26+20\times 0.243=72.26+0.486=72.746\end{array}$

Therefore, the first four sums are $S_1=20,S_2=56,S_3=72.2$ and $S_4=72.746$ .

b.

To find the sum of the series to the 47^th term using the graph.

The sum of the series to the 47^th term is $198.58607$ .

Given information:

The given graph models the sum of the first $n$ terms of the geometric series with $a_1=20$ and $r=0.9$ .

Calculation:

Observe the graph to check that the graph shows that $Y=198.58607$ when $X=47$ .

Therefore, the sum of the series to the 47^th term is $198.58607$ .

c.

To write and evaluate the formula for the sum of the series.

The formula to evaluate the sum of the series is $S=\frac{a_1}{1-r}$ and the sum of the series is $200$ .

Given information:

The given graph models the sum of the first $n$ terms of the geometric series with $a_1=20$ and $r=0.9$ .

Calculation:

The given series is geometric with $a_1=20$ and $r=0.9$ .

Observe that $\left|r\right|=\left|0.9\right|=0.9<1$ , so the series converges. Therefore, the sum of the series can be evaluate using the formula, $S=\frac{a_1}{1-r}$ .

Now, substitute $a_1=20$ and $r=0.9$ in the formula $S=\frac{a_1}{1-r}$ to find the sum.

  $\begin{array}{c}S=\frac{a_1}{1-r}\\ =\frac{20}{1-0.9}\\ =\frac{20}{0.1}\\ =200\end{array}$

Therefore, the sum of the series is 200.

d.

To graph the formula for the sum of the series and verify the answer of part (b).

Given information:

The given graph models the sum of the first $n$ terms of the geometric series with $a_1=20$ and $r=0.9$ .

Calculation:

Use a graphing calculator to graph the sum of the series as shown below.

  

Observe that the graph verifies that the sum of the series to the 47^th term is $198.586$ .