How do you find the first three iterate of the function f(x)=4x-3 for the given initial value x_0=2?

2 Answers
Nov 8, 2016

The value of x_3=3/4

Explanation:

The Newton's method is x_(n+1)=x_n-f(x_n)/(f'(x_n))
Here f(x)=4x-3 and f'(x)=4
And x_0=2
:. x_1=x_0-f(x_0)/(f'(x_0))=2-5/4=3/4

x_2=x_1-f(x_1)/(f'(x_1))=3/4-0=3/4

x_3=x_2-f(x_2)/(f'(x_2))=3/4-0
We obtain the same valuebecause f(x) is a linear function and the intercept with the x-axis is (3/4,0)

Nov 11, 2016

x_1 = 5, x_2 = 17, x_3 = 65

Explanation:

If I understand the question correctly, it is talking about a sequence defined by:

{ (x_0 = 2), (x_n = f(x_(n-1)) = 4x_(n-1)-3 " for " n >= 1) :}

We find:

x_0 = 2

x_1 = 4x_0-3 = 4(2)-3 = 8-3 = 5

x_2 = 4x_1-3 = 4(5)-3 = 20-3 = 17

x_3 = 4x_2-3 = 4(17)-3 = 68-3 = 65

The formula for a general term of the sequence is:

x_n = 4^n+1

as can be proved by induction:

Base case:

x_0 = 2 = 1 + 1 = 4^0 + 1

Induction step:

x_(n+1) = 4x_n - 3 = 4(4^n+1) - 3 = 4^(n+1) + 4 - 3 = 4^(n+1) + 1