Find term or common difference

2 The sum of the first 20 terms of an arithmetic progression is 405 and the sum of the first 40 terms is 1410 . Find the 60th term of the progression.

3 An arithmetic progression has first term 7. The \(n\)th term is 84 and the ( \(3 n\) )th term is 245 .
Find the value of \(n\).

3 The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49 .

1. Find the first term of the progression and the common difference. The \(n\)th term of the progression is 46 .
2. Find the value of \(n\).

3 The 12th term of an arithmetic progression is 17 and the sum of the first 31 terms is 1023. Find the 31st term.

5 In an arithmetic progression the first term is \(a\) and the common difference is 3 . The \(n\)th term is 94 and the sum of the first \(n\) terms is 1420 . Find \(n\) and \(a\).

4. The \(4 ^ { \text {th } }\) term of an arithmetic sequence is 3 and the sum of the first 6 terms is 27 Find the first term and the common difference of this sequence.

8. A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find

1. the number of houses built in 1986, the first year of the building programme,
2. the total number of houses built in the 25 years of the programme.

7. An athlete prepares for a race by completing a practice run on each of 11 consecutive days. On each day after the first day, he runs further than he ran on the previous day. The lengths of his 11 practice runs form an arithmetic sequence with first term \(a \mathrm {~km}\) and common difference \(d \mathrm {~km}\). He runs 9 km on the 11th day, and he runs a total of 77 km over the 11 day period.
Find the value of \(a\) and the value of \(d\).

5. 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

1. the value of \(d\),
2. the value of \(a\),
3. the total number of houses built in Oldtown over the 40-year period.

4. A company, which is making 140 bicycles each week, plans to increase its production. The number of bicycles produced is to be increased by \(d\) each week, starting from 140 in week 1 , to \(140 + d\) in week 2 , to \(140 + 2 d\) in week 3 and so on, until the company is producing 206 in week 12.

1. Find the value of \(d\). After week 12 the company plans to continue making 206 bicycles each week.
2. Find the total number of bicycles that would be made in the first 52 weeks starting from and including week 1.

1 The 20th term of an arithmetic progression is 10 and the 50th term is 70 .

1. Find the first term and the common difference.
2. Show that the sum of the first 29 terms is zero.

7

1. In an arithmetic progression, the first term is 12 and the sum of the first 70 terms is 12915 . Find the common difference.
2. In a geometric progression, the second term is - 4 and the sum to infinity is 9 . Find the common ratio.

8 The 7th term of an arithmetic progression is 6. The sum of the first 10 terms of the progression is 30. Find the 5th term of the progression.

3 The 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term.

1 The 7th term of an arithmetic progression is 6. The sum of the first 10 terms of the progression is 30. Find the 5th term of the progression.

2 The tenth term of an arithmetic progression is equal to twice the fourth term. The twentieth term of the progression is 44 .

1. Find the first term and the common difference.
2. Find the sum of the first 50 terms.

6 An arithmetic progression has first term 7 and third term 12.

1. Find the 20th term of this progression.
2. Find the sum of the 21st to the 50th terms inclusive of this progression.

6 The third term of an arithmetic progression is 24 . The tenth term is 3 .
Find the first term and the common difference. Find also the sum of the 21st to 50th terms inclusive.

10 In an arithmetic progression, the second term is 11 and the sum of the first 40 terms is 3030 . Find the first term and the common difference.

3 An arithmetic progression has tenth term 11.1 and fiftieth term 7.1. Find the first term and the common difference. Find also the sum of the first fifty terms of the progression.

3 The 15th term of an arithmetic sequence is 88. The sum of the first 10 terms is 310 .
Determine the first term and the common difference.

1. A car has six forward gears.

The fastest speed of the car

• in \(1 ^ { \text {st } }\) gear is \(28 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
• in \(6 ^ { \text {th } }\) gear is \(115 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)

Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence,

1. find the fastest speed of the car in \(3 { } ^ { \text {rd } }\) gear. Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,
2. find the fastest speed of the car in \(5 ^ { \text {th } }\) gear.

1. In an arithmetic series

• the first term is 16
• the 21 st term is 24
  1. Find the common difference of the series.
  2. Hence find the sum of the first 500 terms of the series.

5 The first \(n\) terms of an arithmetic series are \(17 + 28 + 39 + \ldots + 281 + 292\).

1. Find the value of \(n\).
2. Find the sum of these \(n\) terms.