Growth & decay

18At the start of year n, the population of deer in a park is D_n At the start of the following year, the population of the deer is D_{n+ 1}

At the start of 2019, the population of the deer was 2000 At the start of 2020, the population of the deer was 2400 Show that, at the start of 2022, the population of the deer was greater than 3000

Peter invests £4500 in a savings account for 3 years. He gets 1.8% per year compound interest. Work out the total amount of interest Peter gets.

Rudi invests £4500 in a savings account. He gets compound interest at a rate of 2.4% for the first year 1.8% for each extra year.

Work out the value of Rudi’s investment at the end of 3 years.

Bruna buys a car for £7500 The value of the car depreciates by x% each year. At the end of 2 years the value of the car is £4107

Work out the value of x.

A company has 25 000 workers. The number of workers increases at a rate of 6% per year for 3 years. Calculate the total number of workers at the end of the 3 years.

A company has 25 000 workers. The number of workers increases at a rate of 6% per year for 3 years. Calculate the total number of workers at the end of the 3 years.

At the start of 2022 Kim invested some money in a savings account. The account paid 3.5% compound interest each year. At the end of 2022 interest was added to the account then Kim took £750 from the account. At the end of 2023 interest was added to the account then Kim took £1000 from the account. There was then £2937.14 in the account. Work out how much money Kim invested at the start of 2022 You must show all your working.

A ball is thrown upwards and reaches a maximum height. The ball then falls and bounces repeatedly. After the nth bounce, the ball reaches a height of h_n After the next bounce, the ball reaches a height given by h_{n+ 1} = 0.55h_n After the 1st bounce, the ball reaches a height of 8 metres. What height does the ball reach after the 4th bounce?

16At the start of year n the population of a species is P_n At the start of the following year the population of the species is given by

The population of the species at the start of year 1 is 8 million. The population of the species at the start of year 2 is 6 million.

Work out the population of the species at the start of year 3

At the start of year 5 the value of k is increased by 0.3 to a new constant value. Louise thinks that from the start of year 5 the population of the species would increase year on year.

Is Louise correct? You must give a reason for your answer.

Andrew invests £4500 in a savings account for 2 years. The account pays compound interest at a rate of 3.4% per year. Calculate how much Andrew has in this savings account at the end of the 2 years.

Andrew invests £4500 in a savings account for 2 years. The account pays compound interest at a rate of 3.4% per year. Calculate how much Andrew has in this savings account at the end of the 2 years.

Tamsin buys a house with a value of £150 000 The value of Tamsin’s house increases by 4% each year. Rachel buys a house with a value of £160 000 The value of Rachel’s house increases by 1.5% each year. At the end of 2 years, whose house has the greater value? You must show how you get your answer.

Tamsin buys a house with a value of £150 000 The value of Tamsin’s house increases by 4% each year. Rachel buys a house with a value of £160 000 The value of Rachel’s house increases by 1.5% each year. At the end of 2 years, whose house has the greater value? You must show how you get your answer.

Ella invests £7000 for 2 years in an account paying compound interest. In the first year, the rate of interest is 3% In the second year, the rate of interest is 1.5% Work out the value of Ella’s investment at the end of 2 years.

Ella invests £7000 for 2 years in an account paying compound interest. In the first year, the rate of interest is 3% In the second year, the rate of interest is 1.5% Work out the value of Ella’s investment at the end of 2 years.

12The number of insects in a population at the start of the year n is P_n The number of insects in the population at the start of year (n + 1) is P_n+1 where

Given that k has a constant value of 1.13

find out how many years it takes for the number of insects in the population to double. You must show how you get your answer.

The value of k actually increases year on year from its value of 1.13 in year 1

How does this affect your answer to part (a)?

A new phone cost £679 The value of the phone decreases at a rate of 4% per year. Work out the value of the phone at the end of 3 years.

A new phone cost £679 The value of the phone decreases at a rate of 4% per year. Work out the value of the phone at the end of 3 years.

Jane bought a new car three years ago. At the end of the first year the value of the car had decreased by 12.5% The value of the car then decreased by 10% each year for the next two years. At the end of the three years, the value of the car was £17 010 Work out the value of the car when Jane bought it three years ago.

The profit made by a shop increases each year. The profit made by the shop in year n is £P_n Given that the profit made by the shop in the next year is £P_{n+ 1} then

The table shows the profit made by the shop in 2018 and in 2019

Work out the profit predicted to be made by the shop in 2021

Louise invests £ x in Better Investments for 3 years. Sadiq invests £x in County Bank for 3 years.

At the end of the 3 years, the value of Louise’s investment is £344 605 Work out the value of Sadiq’s investment at the end of the 3 years.

19At the start of year n, the number of animals in a population is P_n At the start of the following year, the number of animals in the population is P_{n+ 1} where

At the start of 2017 the number of animals in the population was 4000 At the start of 2019 the number of animals in the population was 3610 Find the value of the constant k.

Tariq buys a laptop. He gets a discount of 5% off the normal price. Tariq pays £551 for the laptop.

Work out the normal price of the laptop.

Joan invests £6000 in a savings account. The savings account pays compound interest at a rate of 2.4% for the first year 1.7% for each extra year.

Work out the value of Joan’s investment at the end of 3 years.

A hot air balloon is descending. The height of the balloon n minutes after it starts to descend is h_n metres. The height of the balloon (n +1) minutes after it starts to descend, h_{n+ 1} metres, is given by

The balloon starts to descend from a height of 1200 metres at 09 15 At 09 16 the height of the balloon is 1040 metres. Work out the height of the balloon at 09 18

Sakira invested £3550 in a savings account for 3 years.

She was paid 2.6% per annum compound interest for each of the first 2 years. She was paid R % interest for the third year.

Sakira had £3819.21 in her savings account at the end of the 3 years.

Work out the value of R. Give your answer correct to 1 decimal place.

The number of rabbits on a farm at the end of month n is P_n

At the end of March there are 200 rabbits on the farm.

Work out how many rabbits there will be on the farm at the end of June.

Considering your results in part (a), suggest what will happen to the number of rabbits on the farm after a long time.

Katy invests £200 000 in a savings account for 4 years. The account pays compound interest at a rate of 1.5 % per annum. Calculate the total amount of interest Katy will get at the end of 4 years.

Katy invests £200 000 in a savings account for 4 years. The account pays compound interest at a rate of 1.5 % per annum. Calculate the total amount of interest Katy will get at the end of 4 years.

Northern Bank has two types of account. Both accounts pay compound interest.

Ali invests £2000 in the cash savings account. Ben invests £1600 in the shares account.

Work out who will get the most interest by the end of 3 years. You must show all your working.

In the 3rd year the rate of interest for the shares account is changed to 4% per annum.

Does this affect who will get the most interest by the end of 3 years? Give a reason for your answer.

Northern Bank has two types of account. Both accounts pay compound interest.

Ali invests £2000 in the cash savings account. Ben invests £1600 in the shares account.

Work out who will get the most interest by the end of 3 years. You must show all your working.

In the 3rd year the rate of interest for the shares account is changed to 4% per annum.

Does this affect who will get the most interest by the end of 3 years? Give a reason for your answer.

The number of animals in a population at the start of year t is P_t The number of animals at the start of year 1 is 400

Given that

work out the number of animals at the start of year 3

Jean invests £12 000 in an account paying compound interest for 2 years.

In the first year the rate of interest is x % At the end of the first year the value of Jean’s investment is £12 336 In the second year the rate of interest is %

What is the value of Jean’s investment at the end of 2 years?

At the beginning of 2009, Mr Veale bought a company. The value of the company was 50 000

Each year the value of the company increased by 2%.

Calculate the value of the company at the beginning of 2017 Give your answer correct to the nearest 100

At the beginning of 2009 the value of a different company was 250 000 In 6 years the value of this company increased to 325 000

This is equivalent to an increase of x% each year.

Find the value of x. Give your answer correct to 2 significant figures.

At time t = 0 hours a tank is full of water.

Water leaks from the tank. At the end of every hour there is 2% less water in the tank than at the start of the hour.

The volume of water, in litres, in the tank at time t hours is V_t

Given that

write down the value of k.

Jack bought a new boat for £12 500

The value, £ V, of Jack’s boat at the end of n years is given by the formula

At the end of how many years was the value of Jack’s boat first less than 50% of the value of the boat when it was new?

A savings account pays interest at a rate of R % per year. Jack invests £5500 in the account for one year.

At the end of the year, Jack pays tax on the interest at a rate of 40%. After paying tax, he gets £79.20

Work out the value of R.

Naoby invests £6000 for 5 years. The investment gets compound interest of x % per annum.

At the end of 5 years the investment is worth £8029.35

Work out the value of x.