Chapter 8, Problem 57RE

(b)

To determine

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

  ${\displaystyle \sum _{i=1}^7{5}^{i-1}}$

Answer to Problem 57RE

The sum $19531$ is verified.

Explanation of Solution

Given:

Sequence - ${\displaystyle \sum _{i=1}^7{5}^{i-1}}$

Formula Used:

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

Calculation:

Given sequence: ${\displaystyle \sum _{i=1}^7{5}^{i-1}}$

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

  $\begin{array}{l}r=\frac{5}{1}\\ r=5\end{array}$

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

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

Thus, the graphing table is -

$n$	$S_n=\frac{\left(5^n-1\right)}{4}$
$1$	$1$
$2$	$6$
$3$	$31$
$4$	$156$
$5$	$781$
$6$	$3906$
$7$	$19531$

The sum is $19531$ .

Conclusion:

Hence, the sum $19531$ is verified.