Sure! Let's find the values of [tex]\( x_1 \)[/tex], [tex]\( x_2 \)[/tex], and [tex]\( x_3 \)[/tex] using the recursive formula [tex]\( x_{n+1} = 1 - \frac{3}{x_n^2} \)[/tex] starting with [tex]\( x_0 = -2.5 \)[/tex].
1. Finding [tex]\( x_1 \)[/tex]:
Using the initial value [tex]\( x_0 = -2.5 \)[/tex],
[tex]\[
x_1 = 1 - \frac{3}{x_0^2}
\][/tex]
[tex]\[
x_1 = 1 - \frac{3}{(-2.5)^2}
\][/tex]
[tex]\[
x_1 = 1 - \frac{3}{6.25}
\][/tex]
[tex]\[
x_1 = 1 - 0.48
\][/tex]
[tex]\[
x_1 = 0.52
\][/tex]
2. Finding [tex]\( x_2 \)[/tex]:
Using the value [tex]\( x_1 = 0.52 \)[/tex],
[tex]\[
x_2 = 1 - \frac{3}{x_1^2}
\][/tex]
[tex]\[
x_2 = 1 - \frac{3}{(0.52)^2}
\][/tex]
[tex]\[
x_2 = 1 - \frac{3}{0.2704}
\][/tex]
[tex]\[
x_2 = 1 - 11.094674556213016
\][/tex]
[tex]\[
x_2 = -10.094674556213016
\][/tex]
3. Finding [tex]\( x_3 \)[/tex]:
Using the value [tex]\( x_2 = -10.094674556213016 \)[/tex],
[tex]\[
x_3 = 1 - \frac{3}{x_2^2}
\][/tex]
[tex]\[
x_3 = 1 - \frac{3}{(-10.094674556213016)^2}
\][/tex]
[tex]\[
x_3 = 1 - \frac{3}{102.90465879219869}
\][/tex]
[tex]\[
x_3 = 1 - 0.02943991896746737
\][/tex]
[tex]\[
x_3 = 0.9705600810325326
\][/tex]
So, the values are:
[tex]\[
\begin{array}{l}
x_1 = 0.52 \\
x_2 = -10.094674556213016 \\
x_3 = 0.9705600810325326
\end{array}
\][/tex]
