To find the value of [tex]\( x_3 \)[/tex] using the iterative formula [tex]\( x_{n+1} = 8 + \frac{500}{x_n^2} \)[/tex], we start with the initial value [tex]\( x_1 = 59 \)[/tex].
Step 1: Calculate [tex]\( x_2 \)[/tex]
The formula for [tex]\( x_2 \)[/tex] is:
[tex]\[ x_2 = 8 + \frac{500}{x_1^2} \][/tex]
Substitute [tex]\( x_1 = 59 \)[/tex]:
[tex]\[ x_2 = 8 + \frac{500}{59^2} \][/tex]
Now calculating the square of 59:
[tex]\[ 59^2 = 3481 \][/tex]
Then, calculate the fraction:
[tex]\[ \frac{500}{3481} \approx 0.143636885952313 \][/tex]
Adding this to 8:
[tex]\[ x_2 \approx 8 + 0.143636885952313 \approx 8.143636885952313 \][/tex]
Step 2: Calculate [tex]\( x_3 \)[/tex]
Next, using the value of [tex]\( x_2 \)[/tex] to find [tex]\( x_3 \)[/tex]:
[tex]\[ x_3 = 8 + \frac{500}{x_2^2} \][/tex]
Substitute [tex]\( x_2 \approx 8.143636885952313 \)[/tex]:
[tex]\[ x_3 = 8 + \frac{500}{(8.143636885952313)^2} \][/tex]
Now calculating the square of [tex]\( 8.143636885952313 \)[/tex]:
[tex]\[ (8.143636885952313)^2 \approx 66.32139068154496 \][/tex]
Then, calculate the fraction:
[tex]\[ \frac{500}{66.32139068154496} \approx 7.537396316320297 \][/tex]
Adding this to 8:
[tex]\[ x_3 \approx 8 + 7.537396316320297 \approx 15.537396316320297 \][/tex]
Step 3: Round [tex]\( x_3 \)[/tex]
Finally, round [tex]\( x_3 \)[/tex] to 2 decimal places:
[tex]\[ x_3 \approx 15.54 \][/tex]
So, the value of [tex]\( x_3 \)[/tex] is:
[tex]\[ \boxed{15.54} \][/tex]
