You can use the recursion [tex]x_n=\frac{(2+1)}{2}[/tex] and the process of iteration to estimate the value of [tex]\sqrt{6}[/tex] without using a calculator. What is the value of the 3rd iterate if [tex]x_0=2.4[/tex]? Carry out your answers to the 5th decimal place.
A. [tex]x_3=2.41111[/tex]
B. [tex]x_3=2.4495[/tex]
C. [tex]x_3=2.45111[/tex]
D. [tex]x_3=2.44949[/tex]
Answer (2)
Calculate x 1 using the recursion formula: x 1 = 2 x 0 + x 0 6 = 2.45 .
Calculate x 2 using the recursion formula: x 2 = 2 x 1 + x 1 6 = 2.449489795 .
Calculate x 3 using the recursion formula: x 3 = 2 x 2 + x 2 6 = 2.449489769 .
Round x 3 to the 5th decimal place: 2.44949 .
Explanation
Understanding the Problem We are given the recursive formula x n = f r a c x n − 1 + f r a c 6 x n − 1 2 and the initial value x 0 = 2.4 . Our goal is to find the value of x 3 , rounded to the 5th decimal place. This formula is used to estimate the value of 6 using iteration.
Calculating the Iterates First, we calculate x 1 using the given formula and x 0 = 2.4 :
x 1 = 2 x 0 + x 0 6 = 2 2.4 + 2.4 6 = 2 2.4 + 2.5 = 2 4.9 = 2.45 Next, we calculate x 2 using the formula and the value of x 1 = 2.45 :
x 2 = 2 x 1 + x 1 6 = 2 2.45 + 2.45 6 = 2 2.45 + 2.44897959 = 2 4.89897959 = 2.449489795 Finally, we calculate x 3 using the formula and the value of x 2 = 2.449489795 :
x 3 = 2 x 2 + x 2 6 = 2 2.449489795 + 2.449489795 6 = 2 2.449489795 + 2.449489743 = 2 4.898979538 = 2.449489769 Rounding x 3 to the 5th decimal place, we get x 3 = 2.44949 .
Final Answer Therefore, the value of the 3rd iterate, x 3 , rounded to the 5th decimal place is 2.44949 .
Examples
Estimating square roots through iterative methods has applications in computer graphics for rendering smooth curves and surfaces. By repeatedly refining an initial guess, algorithms can quickly approximate the necessary square root values for lighting calculations and geometric transformations, enhancing the realism and efficiency of visual displays. This technique is also used in control systems for robotics, where precise calculations of distances and angles are essential for accurate movements and navigation.
We calculated the 3rd iterate of the recursion for estimating 6 starting from x 0 = 2.4 . After performing the iterations, we found that x 3 = 2.44949 . Thus, the correct choice is D .2.44949 .
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