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
The value of
Explanation:
The Newton's method is
Here
And
We obtain the same valuebecause
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