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You can use the recursion [tex]x_{n} =\frac{(x_{n-1}+\frac{6}{x_{n-1} } }{2}[/tex] and the process of iteration to estimate the value of [tex]\sqrt6}[/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} =2.44949[/tex]
b. [tex]x_{3} =2.41111[/tex]
c. [tex]x_{3} =2.4495[/tex]
d. [tex]x_{3} =2.45111[/tex]

Sagot :

LammettHash

Use the given recursion and starting value of [tex]x_0 = 2.4[/tex] to find [tex]x_1[/tex] :

[tex]x_1 = \dfrac{x_0 + \frac6{x_0}}2 = \dfrac{2.4 + \frac{6}{2.4}}2 = 2.45[/tex]

Do the same for [tex]x_2[/tex] and [tex]x_3[/tex] :

[tex]x_2 = \dfrac{x_1 + \frac6{x_1}}2 = \dfrac{2.45 + \frac6{2.45}}2 \approx 2.44949[/tex]

[tex]x_3 = \dfrac{x_2+\frac6{x_2}}2 \approx \dfrac{2.44949 + \frac6{2.44949}}2 \approx \boxed{2.44949}[/tex]

(That's not a mistake. This just tells you that the 2nd and 3rd iterates are very close together and have at least the same first 5 digits after the decimal.)

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