Iterative methods for solving equations

f(x) = x³ − 4x − 3

Show that the equation f(x) = 0 has a root between x = 2 and x = 3.

The number of members of a gym is modelled by the rule

u_n+1 = 1.2u_n − 50

where u_n is the number of members after n years.

There are 500 members now, so u₀ = 500.

Work out the number of members the model predicts after 2 years.

A ball is dropped from a height of 5 m onto a hard floor.

After each bounce the ball rises to 60% of the height of the previous bounce, so

h_n+1 = 0.6h_n, where h₀ = 5

1. Work out the height the ball rises to after the third bounce.

2. After which bounce does the ball first rise to less than 1 m?

Amara opens a savings account with £1000.

The account pays 3% interest at the end of each year, and Amara then pays in another £200.

The amount in the account after n years is given by

a_n+1 = 1.03a_n + 200, where a₀ = 1000

Work out the amount in the account after 3 years. Give your answer to the nearest penny.

A sequence is given by the iteration formula

x_n+1 = ³√(5x_n + 2), with x₁ = 2

1. Work out the values of x₂, x₃ and x₄. Give each value correct to 3 decimal places.

2. The values of x_n approach a root of an equation. Write down this equation in the form x³ + ax + b = 0

A sequence is defined by

u_n+1 = ku_n + 3

where k is a constant.

u₁ = 5 and u₂ = 13

1. Find the value of k

2. Find the value of u₄

The number of deer in a woodland is modelled by

p_n+1 = 0.9p_n + 40

where p_n is the number of deer n years from now.

There are 300 deer now, so p₀ = 300.

1. Work out the number of deer the model predicts in 2 years' time.

2. The model predicts that in the long term the number of deer approaches a fixed value L. Work out the value of L.

1. Show that the equation x³ − 7x + 3 = 0 can be rearranged to give

   x = ³√(7x − 3)

2. Using the iteration formula x_n+1 = ³√(7x_n − 3) with x₁ = 2, work out the values of x₂, x₃ and x₄. Give each value correct to 3 decimal places.

f(x) = x² − 6x + 2

1. Show that the equation f(x) = 0 has a root between x = 0 and x = 1

2. Show that the equation x² − 6x + 2 = 0 can be rearranged to give

   x = x² + 26

f(x) = x³ + 4x − 9

1. Show that the equation f(x) = 0 has a root between x = 1 and x = 2

2. Show that the equation x³ + 4x − 9 = 0 can be rearranged to give

   x = ³√(9 − 4x)

3. Starting with x₀ = 1.5, use the iteration formula x_n+1 = ³√(9 − 4x_n) to work out the values of x₁, x₂ and x₃. Give each value correct to 4 decimal places.

f(x) = x² + 2x − 5

1. Show that the equation f(x) = 0 has a root between x = 1 and x = 2

2. Show that the equation x² + 2x − 5 = 0 can be rearranged to give

   x = 5x + 2

3. Starting with x₀ = 1.5, use the iteration formula x_n+1 = 5x_n + 2 to work out the values of x₁, x₂ and x₃. Give each value correct to 4 decimal places.

4. Write down an estimate of this root of f(x) = 0 correct to 2 decimal places.

The equation x³ − 4x − 3 = 0 has a root between 2 and 3.

Use the iteration formula x_n+1 = ³√(4x_n + 3) with x₀ = 2.5.

1. Work out x₁, x₂ and x₃. Give each answer to 4 decimal places.
2. By evaluating f(x) = x³ − 4x − 3 at 2.295 and 2.305, show that the root is 2.30 correct to 2 decimal places.

x³ − 2x² − 1 = 0 has a root between 2 and 3.

1. Show that the equation can be rearranged to give x = 2 + 1x².
2. Use the iteration x_n+1 = 2 + 1x_n² with x₀ = 2 to find x₁, x₂ and x₃, to 4 decimal places.
3. Given that f(2.205) = −0.0033 and f(2.215) = 0.0548 where f(x) = x³ − 2x² − 1, give the root to 2 decimal places.