Chapter 8, Problem 43CR

To determine

To find: The sum of the finite geometric sequence and then use the graphing utility to verify the result.

Answer to Problem 43CR

Thesum of the sequence is $29493$ and it is verified.

Explanation of Solution

Given:

The given sequence is ${\displaystyle \sum _{i=1}^{7}9{\left(-4\right)}^{n-1}}$ .

Calculation:

Consider the given sequence is,

  ${\displaystyle \sum _{i=1}^{7}9{\left(-4\right)}^{n-1}}$

Expand the above equation as,

  $\begin{array}{c}{\displaystyle \sum _{i=1}^{7}9{\left(-4\right)}^{n-1}}=9{\left(-4\right)}^{1-1}+9{\left(-4\right)}^{2-1}+9{\left(-4\right)}^{3-1}\mathrm{...9}{\left(-4\right)}^{7-1}\\ =9+9{\left(-4\right)}^1+9{\left(-4\right)}^2\mathrm{...9}{\left(-4\right)}^6\end{array}$

Consider the sum of finite geometric sequence is of the form,

  $S_n=a_1\left(\frac{1-r^n}{1-r}\right)$

Then,

  $\begin{array}{c}{S}_n=9\left(\frac{1-{\left(\frac{9\left(-4\right)}{9}\right)}^7}{1-\left(\frac{9\left(-4\right)}{9}\right)}\right)\\ =29493\end{array}$

Now check the sum of the sequence by the use of graphing utility as,

In the maple select the summation ${\displaystyle \sum _{i=k}^{n}f}$ from the list of the expression and then substitute the value of $k=1$ , $n=7$ and $f=9{\left(-4\right)}^{n-1}$ .

Thus, the value displayed on the screen is,

  $\begin{array}{l}>{\displaystyle \sum _{i=1}^{7}9{\left(-4\right)}^{i-1}}\\ 29493\\ >\end{array}$

Thus, the sum is verified.