Arithmetic and Geometric progressions - Word problems
Problem 1: In the year 2000 a shop sold 150 computers. Each year the shop sold 10 more computers than
the year before, so that the shop sold 160 computers in 2001, 170 computers in 2002, and so on forming
an arithmetic sequence.
(a) Calculate the total number of computers the shop sold from 2000 to 2013 inclusive.
(b) In the year 2000, the selling price of each computer was £900. The selling price fell by £20 each year,
so that in 2001 the selling price was £880, in 2002 the selling price was £860, and so on forming an
arithmetic sequence.
In a particular year, the selling price of each computer in £s was equal to three times the number of
computers the shop sold in that year.
By forming and solving an equation, find the year in which this occurred.
Problem 2: Peter is driving late at night on a quiet highway. As he enters the ramp, the car starts from rest
and begins accelerating smoothly. Peter, curious about how fast he is going, starts timing the motion with
his phone.
The car’s velocity increases by exactly 3 m/s every second. That means after 1 second, the car is moving
at 3 m/s, after 2 seconds at 6 m/s, and so on. The velocity at each second forms an arithmetic progression
with the common difference is 3 m/s.
(a) Find the car’s velocity after 15 seconds.
𝑣 = 𝑢 + 𝑎𝑡
𝑣𝑛 = 𝑣1 + 𝑑(𝑛 − 1)
(b) When the car reaches a velocity of 60 m/s, Peter drives at a constant speed.
Calculate the distance that the car has traveled by the time it reaches 60 m/s.
𝑆=
𝑛
× (𝑣1 + 𝑣𝑛 )
2
(c) Find after how many seconds Peter has traveled more than 912 meters.
912 = 600 + 312
𝐷 = 𝑆 + 𝑠 ′ => 𝑡𝐷 = 𝑡𝑆 + 𝑡𝑠′
𝑡𝑆 = 20𝑠
𝑠′
𝑡𝑠′ =
= 5.2
60
Problem 3: Sarah is carrying out a series of experiments which involve using increasing amounts of a
chemical. In the first experiment she uses 6 g of the chemical and in the second experiment she uses 7.8 g
of the chemical.
(a) Given that the amounts of the chemical used form an arithmetic progression.
Find the total amount of chemical used in the first 30 experiments.
… => 𝑎1 = 6
… => = 𝑎2 = 7.8
=> 𝑑 = 1.8
𝑛 = 30 => 𝑎30 = …
𝑛
=> ⋯ => 𝑆30 = . (𝑎1 + 𝑎30 ) = …
2
(b) Instead, it is given that the amounts of the chemical used form a geometric progression. Sarah has a
total of 1800 g of the chemical available.
Show that N, the greatest number of experiments possible, satisfies the inequality 𝟏. 𝟑𝑵 ≤ 𝟗𝟏,
and use logarithms to calculate the value of N.
𝑎1 = 6
𝑎2 = 7.8
7.8
=> 𝑟 =
= 1.3
6
𝑇ℎ𝑒 𝑔𝑟𝑒𝑎𝑡𝑒𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑠 = 𝑁
𝑟𝑁 − 1
𝑆𝑁 = 𝑎1 ×
𝑟−1
𝑆𝑎𝑟𝑎ℎ ℎ𝑎𝑠 … 1800 => 𝑆𝑁 ≤ 1800
𝑟𝑁 − 1
=> 𝑎1 ×
≤ 1800
𝑟−1
=> 𝟏. 𝟑𝑵 ≤ 𝟗𝟏
Problem 4: A company predicts a yearly profit of £120 000 in the year 2013. The company predicts that
the yearly profit will rise each year by 5%. The predicted yearly profit forms a geometric sequence with
common ratio of 1.05
(a) Show that the predicted profit in the year 2016 is £138 915.
(b) Find the first year in which the yearly predicted profit exceeds £200 000.
(c) Find the total predicted profit for the years 2013 to 2023 inclusive, giving your answer to the nearest
pound.
Problem 5: A car was purchased for £18000 on 1st January. On 1st January each following year, the value
of the car is 80% of its value on 1st January in the previous year.
(a) Show that the value of the car exactly 3 years after it was purchased is £9216.
(b) The value of the car falls below £1000 for the first time n years after it was purchased.
Find the value of n.
(c) An insurance company has a scheme to cover the maintenance of the car. The cost is £200 for the first
year, and for every following year the cost increases by 12% so that for the 3rd year the cost of the
scheme is £250.88.
Find the cost of the scheme for the 5th year, giving your answer to the nearest penny.
1.00 𝑝𝑜𝑢𝑛𝑑 = 100 𝑝𝑒𝑛𝑛𝑖𝑒𝑠
955.866 𝑝𝑜𝑢𝑛𝑑𝑠 = 95587 𝑝𝑒𝑛𝑛𝑖𝑒𝑠
(d) Find the total cost of the insurance scheme for the first 15 years.