restart;
f := (sqrt(2)-1)^x-0.00000025*(x+1);

plot(f,x=12..17);

df := diff(f, x);

A := x-f/df;

phi := unapply(A, x);

x0 := 17;

x1 := evalf(phi(x0));

x2 := evalf(phi(x1));

x3 := evalf(phi(x2));

x4 := evalf(phi(x3));

x5 := evalf(phi(x4));

x6 := evalf(phi(x5));

x7 := evalf(phi(x6));

x8 := evalf(phi(x7));

x9 := evalf(phi(x8));

x10 := evalf(phi(x9));