Chapter 8, Problem 59RE

(b)

To determine

To determine the sum of the Geometric sequence using the table feature of the graphing utility -

  ${\displaystyle \sum _{n=1}^{7}3{\left(-4\right)}^{n-1}}$

Answer to Problem 59RE

The sum $9831$ is verified.

Explanation of Solution

Given:

Sequence - ${\displaystyle \sum _{n=1}^{7}3{\left(-4\right)}^{n-1}}$

Formula Used:

Sum of the Geometric sequence is given as $S_n=\frac{a\left(r^n-1\right)}{r-1}$

Calculation:

Given sequence: ${\displaystyle \sum _{n=1}^{7}3{\left(-4\right)}^{n-1}}$

Common difference $=$ $r=\frac{a_2}{a_1}$

  $\begin{array}{l}r=-\frac{12}{3}\\ r=-4\end{array}$

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

  $\begin{array}{l}{S}_n=\frac{3\left({\left(-4\right)}^n-1\right)}{-4-1}\\ S_n=-\frac{3}{5}\left({\left(-4\right)}^n-1\right)\end{array}$

Thus, the graphing table is -

$n$	$S_n=-\frac{3}{5}\left({\left(-4\right)}^n-1\right)$
$1$	$3$
$2$	$-9$
$3$	$39$
$4$	$-153$
$5$	$615$
$6$	$-2457$
$7$	$9831$

The sum is $9831$ .

Conclusion:

Hence, the sum $9831$ is verified.