To find the value of [tex]\( x_3 \)[/tex] using the given iterative formula, let's follow each step carefully:
1. Starting Value: [tex]\( x_1 = 9 \)[/tex].
2. Calculate [tex]\( x_2 \)[/tex] using the formula [tex]\( x_{n+1} = \frac{14}{x_n^2 - 2} \)[/tex]:
[tex]\[
x_2 = \frac{14}{x_1^2 - 2}
\][/tex]
Substituting [tex]\( x_1 = 9 \)[/tex] into the formula:
[tex]\[
x_2 = \frac{14}{9^2 - 2} = \frac{14}{81 - 2} = \frac{14}{79} \approx 0.17721518987341772
\][/tex]
3. Calculate [tex]\( x_3 \)[/tex] using the formula [tex]\( x_{n+1} = \frac{14}{x_n^2 - 2} \)[/tex]:
[tex]\[
x_3 = \frac{14}{x_2^2 - 2}
\][/tex]
Substituting [tex]\( x_2 \approx 0.17721518987341772 \)[/tex] into the formula:
[tex]\[
x_3 = \frac{14}{(0.17721518987341772)^2 - 2}
\][/tex]
Find [tex]\( 0.17721518987341772^2 \)[/tex]:
[tex]\[
(0.17721518987341772)^2 \approx 0.031413611967418683
\][/tex]
Now, substitute this value back into the formula for [tex]\( x_3 \)[/tex]:
[tex]\[
x_3 = \frac{14}{0.031413611967418683 - 2} = \frac{14}{-1.968586388032581317} \approx -7.111671821585545
\][/tex]
4. Round [tex]\( x_3 \)[/tex] to 2 decimal places:
[tex]\[
x_3 \approx -7.11
\][/tex]
Therefore, the value of [tex]\( x_3 \)[/tex] to 2 decimal places is [tex]\(-7.11\)[/tex].
