1.04h Arithmetic sequences: nth term and sum formulae

Consider the binomial expansion of \(\left(1 + \frac{x}{5}\right)^n\) in ascending powers of \(x\), where \(n\) is a positive integer.

1. Write down the first four terms of the expansion, giving the coefficients as polynomials in \(n\). [1]

The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.

1. Show that \(n^3 - 33n^2 + 182n = 0\). [3]
2. Hence find the possible values of \(n\) and the corresponding values of the common difference. [3]

A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_1 = 7\) and \(u_{n+1} = u_n + 4\) for \(n \geq 1\).

1. State what type of sequence this is. [1]
2. Find \(u_{17}\). [2]

A car has six forward gears. The fastest speed of the car • in 1st gear is 28 km h⁻¹ • in 6th gear is 115 km 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 3rd gear. [3]

Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,

1. find the fastest speed of the car in 5th gear. [3]

In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g_n\) and the \(n\)th term of an arithmetic progression is denoted by \(a_n\). It is given that \(g_1 = a_1 = 1 + \sqrt{5}\), \(g_2 = a_2\) and \(g_3 + a_3 = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2\sqrt{5}\). [12]

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, [5]
2. the total number of houses built in the 25 years of the programme. [2]

A small company which makes batteries for electric cars has a 10 year plan for growth. In year 1 the company will make 2600 batteries. In year 10 the company aims to make 12000 batteries. In order to calculate the number of batteries it will need to make each year from year 2 to year 9, the company considers two models. Model A assumes that the number of batteries it will make each year will increase by the same number each year.

1. According to model A, determine the number of batteries the company will make in year 2. Give your answer to the nearest whole number of batteries. [3]

Model B assumes that the numbers of batteries it will make each year will increase by the same percentage each year.

1. According to model B, determine the number of batteries the company will make in year 2. Give your answer to the nearest 10 batteries. [3]

Sam calculates the total number of batteries made from year 1 to year 10 inclusive, using each of the two models.

1. Calculate the difference between the two totals, giving your answer to the nearest 100 batteries. [3]

An arithmetic series has first term \(a\) and common difference \(d\). Given that the sum of the first 9 terms is 54

1. show that $$a + 4d = 6$$ [2]

Given also that the 8th term is half the 7th term,

1. find the values of \(a\) and \(d\). [4]

A sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 8 \quad \text{and} \quad u_{n+1} = u_n + 3.$$

1. Show that \(u_5 = 20\). [1]
2. The \(n\)th term of the sequence can be written in the form \(u_n = pn + q\). State the values of \(p\) and \(q\). [2]
3. State what type of sequence it is. [1]
4. Find the value of \(N\) such that \(\sum_{n=1}^{2N} u_n - \sum_{n=1}^{N} u_n = 1256\). [5]

The 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term. [5]

The first three terms of an arithmetic series are \(9p\), \(8p - 3\), \(5p\) respectively, where \(p\) is a constant. Given that the sum of the first \(n\) terms of this series is \(-1512\), find the value of \(n\). [6]

An arithmetic progression has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero. The first, third and fourth terms of the arithmetic progression are consecutive terms of a geometric progression with common ratio \(r\).

1. 1. Show that \(r = \frac{a + 2d}{a}\). [1]
   2. Find \(d\) in terms of \(a\). [2]
2. Find the common ratio of the geometric progression. [2]

An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 29 terms is 1102.

1. Show that \(a + 14d = 38\). (3 marks)
2. The sum of the second term and the seventh term is 13. Find the value of \(a\) and the value of \(d\). (4 marks)

The first three terms of an arithmetic series are \(9p\), \(8p - 3\), \(5p\) respectively, where \(p\) is a constant. Given that the sum of the first \(n\) terms of this series is \(-1512\), find the value of \(n\). [6]

Business A made a £5000 profit during its first year. In each subsequent year, the profit increased by £1500 so that the profit was £6500 during the second year, £8000 during the third year and so on. Business B made a £5000 profit during its first year. In each subsequent year, the profit was 90% of the previous year's profit.

1. Find an expression for the total profit made by business A during the first \(n\) years. Give your answer in its simplest form. [2]
2. Find an expression for the total profit made by business B during the first \(n\) years. Give your answer in its simplest form. [3]
3. Find how many years it will take for the total profit of business A to reach £385 000. [3]
4. Comment on the profits made by each business in the long term. [2]

An arithmetic progression has 13th term equal to 60 and 31st term equal to 141.

1. Find the first term and common difference of the progression. [3]

A second arithmetic progression has first term 1.5 and common difference 3.

1. 1. Write down the first four terms of each progression. [1]
   2. Prove that the two progressions have an infinite number of terms in common. [2]

An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).

1. Find \(d\) in terms of \(a\) and show that \(r = \frac{5}{3}\). [7]
2. Find the sum to infinity of the geometric progression in terms of \(a\). [2]