Question

PLEASE HELP!

You can use the recursion
`x_(n) =((x_(n-1)+(6)/(x_(n-1) ) )/(2)` and the process of iteration to estimate the value of
`\sqrt6}` without using a calculator. What is the value of the 3rd iterate if
`x_(0) =2.4`? Carry out your answers to the 5th decimal place.
a.
`x_(3) =2.44949`
b.
`x_(3) =2.41111`
c.
`x_(3) =2.4495`
d.
`x_(3) =2.45111`

Answer 1

Use the given recursion and starting value of
`x_0 = 2.4` to find
`x_1` :

`x_1 = \frac{x_0 + \frac6{x_0}}2 = \frac{2.4 + (6)/(2.4)}2 = 2.45`

Do the same for
`x_2` and
`x_3` :

`x_2 = \frac{x_1 + \frac6{x_1}}2 = \frac{2.45 + \frac6{2.45}}2 \approx 2.44949`

`x_3 = \frac{x_2+\frac6{x_2}}2 \approx \frac{2.44949 + \frac6{2.44949}}2 \approx \boxed{2.44949}`

(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.)