Net-Zero Buses

15 cities, each of varying sizes, decide to have carbon-neutral public transport systems.

When the cities are arranged in size order from smallest to largest, the number of new electric buses they need follows an arithmetic sequence with first term (representing the smallest amount of buses needed for the smallest city) a buses and common difference d buses. 

The 7ᵗʰ city in this progression needs 580 new electric buses and the largest city needs 1020 new electric buses.

Find the value of a and the value of d.

[5 marks]

Arithmetic Afforestation

A country decides to begin a reforestation program, starting in 2020, gradually increasing the number of trees planted per year by the same amount each year.

The timetable for the first four years is shown below:

Year: 2020 2021 2022 2023
Trees planted: 6.0×10 1.4×10 2.2×10 3.0×10

a) Find an expression, in terms of n for the number of trees planted in year n.

[2 marks]

b) Calculate how many trees in total will be planted if this program is followed for 10 years.

[2 mark] 

The government will declare the program a success once 2.45×10 trees have been planted in total

c) Given that the country plants all of the trees that the model would predict in year k, but reaches the target part way through year (k + 1), show that k satisfies 

(2k – 49)(k + 25) < 0.

[5 marks]

EV Increase

A country decides to subsidise the purchase of electric vehicles, causing more people to buy them.

Initially, the country used an equivalent of 56 million tonnes of oil for transport.

Since the government subsidy, the transport sector in each subsequent year has required 80% of the amount of oil that the previous year required.


a) Write a recurrence relation to model the amount of oil used, in million tonnes, in each subsequent year.

[2 marks]

b) Find the amount of oil needed for the transport sector after the fifth year.

[2 marks]

c) Find the total amount of oil used until the transport sector requires no more oil.

[4 marks]

d) State one limitation with the model.

[1 mark]

A Total Emissions Goal

The warming caused by carbon dioxide (CO₂) emissions over any given period is proportional to the total amount of CO₂ emitted over that period. Recognising this, a country decides to limit its carbon dioxide emissions to less than 12×10¹²kg in total emitted across 20 years.

For the first 4 years, the countries emissions are stable at 800×10⁹kg year⁻¹

The country decides that they will be able to reduce their emissions so that each subsequent year produces 5% less emissions than the previous year.

Using the model,

a) Show that the country’s total CO₂ emissions from the
first 6 years are estimated to be 4682×10⁹kg CO₂

[2 marks]

Show that the estimated total emissions per year in the rᵗʰ year, with units
x10⁹kg year⁻¹, for 5 ≤ r ≤ 20, is

800 x 0.95ʳ⁻⁴

[1 mark]

Determine whether the country will meet their 20 year emissions goal.

[4 marks]