3. of 89

6. Each repetition of the process is also called an "iteration”.
Iteration means the act of REPEATING A PROCESS with the aim of approaching a desired goal, target or result.
Each repetition of the process is also called an "iteration”.
Means the next number in the sequence (n+1) is four times the previous term (n), add one
xn+1 = 4xn + 1

8. Rearrange to isolate one of the xs on the LHS
Show that two possible rearrangements of x3 – 4x + 1 = 0 lead to the iterative formulae
xn+1 = ¼(xn3 + 1) and xn+1 = 3√(4xn – 1)
Rearrange to isolate one of the xs on the LHS
x3 – 4x + 1 = 0
4x = x3 + 1
x = ¼(x3 + 1)
xn+1 = ¼ (xn3 + 1)
x3 – 4x + 1 = 0
x3 = 4x - 1
x = 3√4x - 1
xn+1 = 3√4xn – 1

9. Find the positive roots of x3 – 4x + 1 = 0, correct to 3 decimal places, by putting x0 = 1 in xn+1 = ¼(xn3 + 1)
xn+1 = ¼(xn3 + 1)
x0 = 1
x1 = ¼ (13 + 1) = 0.5
x2 = ¼ (ANS3 + 1) = …
x3 = Just press = …
x4 = …
x5 = …
Always write the calculator display to a couple more decimal places than asked for in the question
The sequence converges and the answer can be rounded to (3dp).

10. This sequence diverges.
Show that putting in x0 = 2 in xn+1 = ¼(xn3 + 1) leads to a divergent sequence
xn+1 = xn3 + 1 4
x0 = 2
x1 = 2.25
x2 = 3.097…
x3 = 7.680…
x4 = …
This sequence diverges.

11. Show that 𝑥 2 −6𝑥+2=0 can be written in the form 𝑥 = 6𝑥 −2
Use the iteration formula 𝑥 𝑛+1 = 6 𝑥 𝑛 −2 to find 𝑥 4 to 2 decimal places. Start with 𝑥 0 =2.
𝑥 𝑛+1 = 6 𝑥 𝑛 −2
𝑥 2 −6𝑥+2=0
𝑥 1 = 6(2) −2 =3.1622…
𝑥 2 =6𝑥 −2
𝑥 2 =4.1199…
𝑥= 6𝑥 −2
𝑥 3 =4.7664…
𝑥 4 =5.1574…=5.16 (2 dp)

12. TRY YOURSELF NOW Answers 1) 5.40 5) 0.38 2) 2.79 6) 1.82
1) ) 0.38
2) ) 1.82
3) ) 1.5
4) ) 12.81