Arithmetic sequences and series

The first three terms of an arithmetic sequence are 6 k, 10 and 2 k where k is a constant.

Find the value of k.

Hence find the value of the sum of the first 50 terms of this sequence.

Jamie takes out an interest‑free loan of £8100

Jamie makes a payment every month to pay back the loan.

Jamie repays £400 in month 1, £390 in month 2, £380 in month 3, and so on, so that the amounts repaid each month form an arithmetic sequence.

Show that Jamie repays £290 in month 12

After Jamie’s Nth payment, the loan is completely paid back.

Show thatN² – 81N + 1620 = 0

Hence find the value of N.

Express 2 cosθ + 8 sinθ in the form Rcos(θ – α), where R and α are constants,

The first three terms of an arithmetic sequence are

Given that S₉ represents the sum of the first 9 terms of this sequence as x varies,

find the exact maximum value of S₉

deduce the smallest positive value of x at which this maximum value

In an arithmetic series

• the first term is 16

• the 21st term is 24

Find the common difference of the series.

Hence find the sum of the first 500 terms of the series.

In an arithmetic series

• the first term is 16

• the 21st term is 24

Find the common difference of the series.

Hence find the sum of the first 500 terms of the series.

A car has six forward gears. The fastest speed of the car ● in 1^st gear is 28 km h^–1 ● in 6^th gear is 115 km h^–1 Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence,

find the fastest speed of the car in 3^rd gear. Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,

find the fastest speed of the car in 5^th gear.

A sequence u₁, u₂, u₃, … is defined by

Jess started work 20 years ago. In year 1 her annual salary was £17000. Her annual salary increased by £1500 each year, so that her annual salary in year 2 was £18500, in year 3 it was £20000 and so on, forming an arithmetic sequence. This continued until she reached her maximum annual salary of £32000 in year k. Her annual salary then remained at £32000.

Find the value of the constant k.

Calculate the total amount that Jess has earned in the 20 years.

Xin has been given a 14 day training schedule by her coach. Xin will run for A minutes on day 1, where A is a constant. She will then increase her running time by (d + 1) minutes each day, where d is a constant.

Show that on day 14, Xin will run for (A + 13 d + 13) minutes.

Yi has also been given a 14 day training schedule by her coach. Yi will run for (A – 13) minutes on day 1. She will then increase her running time by (2 d – 1) minutes each day. Given that Yi and Xin will run for the same length of time on day 14,

find the value of d.

Given that Xin runs for a total time of 784 minutes over the 14 days,

find the value of A.

A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week N.

Find the value of N.

The company then plans to continue to make 600 mobile phones each week.

Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.

Each year, Abbie pays into a savings scheme. In the first year she pays in £500. Her payments then increase by £200 each year so that she pays £700 in the second year, £900 in the third year and so on.

Find out how much Abbie pays into the savings scheme in the tenth year. (2) Abbie pays into the scheme for n years until she has paid in a total of £67200.

Hence find the number of years that Abbie pays into the savings scheme. (2)

Lewis played a game of space invaders. He scored points for each spaceship that he captured. Lewis scored 140 points for capturing his first spaceship. He scored 160 points for capturing his second spaceship, 180 points for capturing his third spaceship, and so on. The number of points scored for capturing each successive spaceship formed an arithmetic sequence.

Find the number of points that Lewis scored for capturing his 20th spaceship.

Find the total number of points Lewis scored for capturing his first 20 spaceships.

Sian played an adventure game. She scored points for each dragon that she captured. The number of points that Sian scored for capturing each successive dragon formed an arithmetic sequence. Sian captured n dragons and the total number of points that she scored for capturing all n dragons was 8500. Given that Sian scored 300 points for capturing her first dragon and then 700 points for capturing her n th dragon,

find the value of n.

A boy saves some money over a period of 60 weeks. He saves 10p in week 1, 15p in week 2, 20p in week 3 and so on until week 60. His weekly savings form an arithmetic sequence.

Find how much he saves in week 15

Calculate the total amount he saves over the 60 week period.

The boy’s sister also saves some money each week over a period of m weeks. She saves 10p in week 1, 20p in week 2, 30p in week 3 and so on so that her weekly savings form an arithmetic sequence. She saves a total of £63 in the m weeks.

Show that m (m + 1) = 35 × 36

Hence write down the value of m.

A farmer has a pay scheme to keep fruit pickers working throughout the 30 day season. He pays £ a for their first day, £(a + d) for their second day, £(a + 2 d) for their third day, and so on, thus increasing the daily payment by £ d for each extra day they work. A picker who works for all 30 days will earn £40.75 on the final day.

Use this information to form an equation in a and d.

A picker who works for all 30 days will earn a total of £1005

Show that 15(a + 40.75) = 1005

Hence find the value of a and the value of d.

Jill gave money to a charity over a 20-year period, from Year 1 to Year 20 inclusive. She gave £150 in Year 1, £160 in Year 2, £170 in Year 3, and so on, so that the amounts of money she gave each year formed an arithmetic sequence.

Find the amount of money she gave in Year 10.

Calculate the total amount of money she gave over the 20-year period.

Kevin also gave money to the charity over the same 20-year period. He gave £ A in Year 1 and the amounts of money he gave each year increased, forming an arithmetic sequence with common difference £30. The total amount of money that Kevin gave over the 20-year period was twice the total amount of money that Jill gave.

Calculate the value of A.

A 40-year building programme for new houses began in Oldtown in the year 1951 (Year 1) and finished in 1990 (Year 40). The numbers of houses built each year form an arithmetic sequence with first term a and common difference d. Given that 2400 new houses were built in 1960 and 600 new houses were built in 1990, find

the value of d,

the value of a,

the total number of houses built in Oldtown over the 40-year period.

The first term of an arithmetic series is a and the common difference is d. The 18th term of the series is 25 and the 21st term of the series is 3232

Use this information to write down two equations for a and d.

Show that a = –17.5 and find the value of d.

The sum of the first n terms of the series is 2750.

Show that n is given by

Hence find the value of n.

Sue is training for a marathon. Her training includes a run every Saturday starting with a run of 5 km on the first Saturday. Each Saturday she increases the length of her run from the previous Saturday by 2 km.

Show that on the 4th Saturday of training she runs 11 km.

Find an expression, in terms of n, for the length of her training run on the n th Saturday.

Show that the total distance she runs on Saturdays in n weeks of training is n (n + 4) km.

On the n th Saturday Sue runs 43 km.

Find the value of n.

Find the total distance, in km, Sue runs on Saturdays in n weeks of training.

The first term of an arithmetic sequence is 30 and the common difference is –1.5

Find the value of the 25th term.

The r th term of the sequence is 0.

Find the value of r.

The sum of the first n terms of the sequence is S_n .

Find the largest positive value of S_n .