- Some equations do not have “nice” solutions
- They are
**not**integers (whole numbers), fractions or simple decimals - Instead, they can be irrational decimal solutions that go on forever with no pattern

- They are
**Iteration**is a repeated process used to solve such equations- the process starts with an
**initial****value**(starting value) - after each stage of the process (after each "iteration"), a solution is produced
- the solutions get more and more accurate as more and more iterations are performed

- these solutions are called
**estimates**

- these solutions are called

- the process starts with an
- Scientific calculators allow us to perform iterations very quickly using the ANS button
- Iteration questions will only be asked in the calculator exam

- Find the equation you would like to solve using iteration
- for example, x
^{3}+*x*=

- for example, x
- Rearrange this equation into the form
*x*= f(*x*) by making any*x*the subject of the equation- for example,

- Replace the
*x*on the left with*x*_{n+1}(meaning the "next" value of*x*) and any*x*'s on the right with*x*(meaning the "current" value of_{n }*x*)- This called the
**iterative formula**and**n**are just counters:**n+1**is simply one more than**n+1****n****n**starts at**0**so the process starts with**x**_{0}_{,}the initial value_{ }*x*_{1}is your first estimate,*x*_{2}is your second estimate, etc

- Find a good
**initial (starting) value (**near to the solution*x*_{0})- This is often given in the question, for example
*x*_{0}= 2

- This is often given in the question, for example
- Store
*x*_{0}= 2 into your calculator (by typing 2 and pressing the "=" button)- 2 is now stored under the "Ans" button

- Type in the right-hand side of the iteration formula with "Ans" instead of
*x*_{n} - Press "=" to find
*x*_{1}(be careful to only press "=" once)*x*_{1}= 1.709975...

- Without pressing any other button, press "=" again to find
*x*_{2}*x*_{2}= 1.742418...

- Press "=" again to find
*x*_{3}*x*_{3}= 1.738849...

- Repeat as many times as required
- the more you do, the closer the estimates get to the true solution

*x*_{1},*x*_{2},*x*_{3}... etc are**estimates**to the**solution**of*x*= f(*x*)- for example,
*x*_{1}= 1.709975...,*x*_{2}= 1.742418...,*x*_{3}= 1.738849... are estimates to

- for example,
- They are also estimates of solutions to any
**rearrangements**of*x*= f(*x*)- such as the original equation trying to be solved,
*x*^{3}+*x*=

- such as the original equation trying to be solved,
- This makes
*x*_{1},*x*_{2},*x*_{3}... estimates to the solution of the original equation - The more times you perform the iteration, the better the estimates get to the real solution

- To find x
_{0 }(the initial / starting value), you are often asked to show that there is a solution in an interval - For example, show that there is a solution to
*x*^{3}+*x*=- Method 1: Leave a constant term (e.g. the 7) on the right, substitute
*x*= 1 and*x*= 2 into the left and show that this gives values**below and above**7- 1
^{3}+ 1 = 2 and 2^{3}+ 2 = 10 which are below and above 7 - A solution therefore lies between 1 and 2

- 1
- Method 2: Use "0" as your constant term on the right (by rearranging the equation into "... = 0"), then substitute in
*x*= 1 and x = 2*,*showing this gives values below and above 0, i.e.**negative and positive** - this is called a
**change of sign**between 1 and 2*x*^{3}+*x*-- Substitute x = 1 into the left-hand side: 1
^{3}+ 1 - 7 = -5 (negative) - Substitute x = 2 into the left-hand side: 2
^{3}+ 2 - 7 = 3 (positive) - A solution lies between 1 and 2 as there is a
**change of sign**

- Method 1: Leave a constant term (e.g. the 7) on the right, substitute
- Knowing an interval that contains the solution helps to find x
_{0}- If the solution is between 1 and 2 then you could choose either
*x*_{0}= 1 or*x*_{0}= 2

- If the solution is between 1 and 2 then you could choose either

- Be careful to not press =/EXE or "Ans" more than once at a time. If you do the best thing to do is to restart from the beginning.
- Iteration questions always require working with a lot of decimal places, so write down all digits from your calculator display for
*x*_{1},*x*_{2}, etc. and round them at the end if necessary

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