**Using an iterative formula**

Activity

Solve the iteration questions.

Example

(a) Show that the equation 2x -3x -2x -1 = 0 has a solution between 1 < x < 2.

(b) The formula x = can be used to solve the equation

above. Use x = 1 and find the values of x , x , x , x , and x . Hence give the value of x to 2 decimal places.

**Understand what an iterative formula is, the form it takes and how it is used to solve equations.**

Starter

(a) Show that the equation x - 2x + x - 3 = 0 has a solution in the range 2 < x < 3.

x = 2 gives -1, x = 3 gives 9.

(b) Substitute the value of 2 into the expression

2.2

(c) Substitute the answer you got to (b) into the same expression.

2.175

(d) Keep substituting in the previous answer you got into the expression. What happens?

Eventually settles down to 2.17455941

3

2

2x - 2x + 3

3x - 4x +1

3

2

2

Starter

(a) Show that the equation x - 2x + x - 3 = 0 has a solution in the range 2 < x < 3.

(b) Substitute the value of 2 into the expression

(c) Substitute the answer you got to (b) into the same expression.

(d) Keep substituting in the previous answer you got into the expression. What happens?

3

2

2x - 2x + 3

3x - 4x +1

3

2

2

Iterative formulae

The formula x = can be used to solve the equation

x - 2x + x - 3 = 0.

This is an example of an iterative formula.

An iterative formula works by taking a starting value that is close to the solution to the equation, and generating a new value that is closer. It then takes the closer value and generates a new value that is even closer. This process continues until a solution that is accurate enough is generated.

2x - 2x + 3

3x - 4x +1

3

2

2

n+1

n

n

n

n

3

2

Iterative formulae

The formula x = can be used to solve the equation

x - 2x + x - 3 = 0.

Use the value of x = 2. Write down the values of x , x , x , and x .

Hence give the solution to the equation x -2x + x - 3= 0 to 3 s.f.

2x - 2x + 3

3x - 4x +1

3

2

2

n+1

n

n

n

n

3

2

3

3

2

2

1

0

4

Iterative formulae

The formula x = can be used to solve the equation

x - 2x + x - 3 = 0.

Use the value of x = 2. Write down the values of x , x , x , and x .

Hence give the solution to the equation x -2x + x - 3= 0 to 3 s.f.

2x - 2x + 3

3x - 4x +1

3

2

2

n+1

n

n

n

n

3

2

3

3

2

2

1

0

4

x = 2.2

x = 2.175

x = 2.17455...

x = 2.17455...

so x = 2.17 to 3 s.f.

1

2

3

4

4

3

3

4

2

1

0

5

n+1

3x + 2x + 1

2x

3

n

n

n

2

Example

(a) Show that the equation 2x -3x -2x -1 = 0 has a solution between 1 < x < 2.

x = 1 gives -4 and x = 2 gives 3. The change of sign indicates a solution.

(b) The formula x = can be used to solve the equation

above. Use x = 1 and find the values of x , x , x , x , and x . Hence give the value of x to 2 decimal places.

x = 1.73205...

x = 1.828139...

x = 1.85440...

x = 1.861790...

x = 1.86388.....

4

3

3

4

2

1

0

5

n+1

3x + 2x + 1

2x

3

n

n

n

2

x = 1.86 (2 d.p.)

1

2

3

4

5

Activity

Solve the iteration questions.

1) 5.61

2) 2.79

3) 3.74

4) 6.23

5) 1.14

6) 1.82

7) -2.43

8) 1.22

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