1.04h Arithmetic sequences: nth term and sum formulae

Ahmed plans to save £250 in the year 2001, £300 in 2002, £350 in 2003, and so on until the year 2020. His planned savings form an arithmetic sequence with common difference £50.

1. Find the amount he plans to save in the year 2011. [2]
2. Calculate his total planned savings over the 20 year period from 2001 to 2020. [3]

Ben also plans to save money over the same 20 year period. He saves £\(A\) in the year 2001 and his planned yearly savings form an arithmetic sequence with common difference £60. Given that Ben's total planned savings over the 20 year period are equal to Ahmed's total planned savings over the same period,

1. calculate the value of \(A\). [4]

1. An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is $$\frac{1}{2}n[2a + (n - 1)d].$$ [4]

A company made a profit of £54 000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference £\(d\). This model predicts total profits of £619 200 for the 9 years 2001 to 2009 inclusive.

1. Find the value of \(d\). [4]

Using your value of \(d\),

1. find the predicted profit for the year 2011. [2]

An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06. Using this alternative model and again taking the profit in 2001 to be £54 000,

1. find the predicted profit for the year 2011. [3]

Shelim starts his new job on a salary of £14000. He will receive a rise of £1500 a year for each full year that he works, so that he will have a salary of £15500 in year 2, a salary of £17000 in year 3 and so on. When Shelim's salary reaches £26000, he will receive no more rises. His salary will remain at £26000.

1. Show that Shelim will have a salary of £26000 in year 9. [2]
2. Find the total amount that Shelim will earn in his job in the first 9 years. [2]

Anna starts her new job at the same time as Shelim on a salary of £\(A\). She receives a rise of £1000 a year for each full year that she works, so that she has a salary of £\((A + 1000)\) in year 2, £\((A + 2000)\) in year 3 and so on. The maximum salary for her job, which is reached in year 10, is also £26000.

1. Find the difference in the total amount earned by Shelim and Anna in the first 10 years. [6]

The first three terms of an arithmetic series are \(p\), \(5p - 8\), and \(3p + 8\) respectively.

1. Show that \(p = 4\). [2]
2. Find the value of the 40th term of this series. [3]

In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by \(x\) phones per month for the next 35 months, so that \((280 + x)\) phones will be sold in the second month, \((280 + 2x)\) in the third month, and so on. Using this model with \(x = 5\), calculate

1. 1. the number of phones sold in the 36th month, [2]
   2. the total number of phones sold over the 36 months. [2]

The shop sets a sales target of 17000 phones to be sold over the 36 months. Using the same model,

1. find the least value of \(x\) required to achieve this target. [4]

Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays £500. Her payments then increase by £50 each year, so that she pays £550 in the second year, £600 in the third year, and so on.

1. Find the amount that Anne will pay in the 40th year. [2]
2. Find the total amount that Anne will pay in over the 40 years. [2]

Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in £890 and his payments then increase by £\(d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,

1. find the value of \(d\). [4]

Evaluate $$\sum_{r=10}^{30} (7 + 2r).$$ [4]

A sequence is defined by the recurrence relation $$u_{n+1} = u_n - 2, \quad n > 0, \quad u_1 = 50.$$

1. Write down the first four terms of the sequence. [1]
2. Evaluate $$\sum_{r=1}^{20} u_r.$$ [3]

The first three terms of an arithmetic series are \((12 - p)\), \(2p\) and \((4p - 5)\) respectively, where \(p\) is a constant.

1. Find the value of \(p\). [2]
2. Show that the sixth term of the series is 50. [3]
3. Find the sum of the first 15 terms of the series. [2]
4. Find how many terms of the series have a value of less than 400. [3]

A store begins to stock a new range of DVD players and achieves sales of £1500 of these products during the first month. In a model it is assumed that sales will decrease by £\(x\) in each subsequent month, so that sales of £\((1500 - x)\) and £\((1500 - 2x)\) will be achieved in the second and third months respectively. Given that sales total £8100 during the first six months, use the model to

1. find the value of \(x\), [4]
2. find the expected value of sales in the eighth month, [2]
3. show that the expected total of sales in pounds during the first \(n\) months is given by \(kn(51 - n)\), where \(k\) is an integer to be found. [3]
4. Explain why this model cannot be valid over a long period of time. [1]

The third term of an arithmetic series is \(5\frac{1}{2}\). The sum of the first four terms of the series is \(22\frac{3}{4}\).

1. Show that the first term of the series is \(6\frac{1}{4}\) and find the common difference. [7]
2. Find the number of positive terms in the series. [3]
3. Hence, find the greatest value of the sum of the first \(n\) terms of the series. [2]

The first two terms of an arithmetic series are \((t - 1)\) and \((t^2 - 5)\) respectively, where \(t\) is a positive constant.

1. Find and simplify expressions in terms of \(t\) for
   1. the common difference of the series,
   2. the third term of the series. [4]

Given also that the third term of the series is 19,

1. find the value of \(t\), [2]
2. show that the 10th term of the series is 75, [3]
3. find the sum of the first 40 terms of the series. [2]

A sequence \(S\) has terms \(u_1, u_2, u_3, \ldots\) defined by $$u_n = 3n - 1,$$ for \(n \geqslant 1\).

1. Write down the values of \(u_1, u_2\) and \(u_3\), and state what type of sequence \(S\) is. [3]
2. Evaluate \(\sum_{n=1}^{100} u_n\). [3]

In an arithmetic progression the first term is 15 and the twentieth term is 72. Find the sum of the first 100 terms. [4]

A sequence is given by the following. $$u_1 = 3$$ $$u_{n+1} = u_n + 5$$

1. Write down the first 4 terms of this sequence. [1]
2. Find the sum of the 51st to the 100th terms, inclusive, of the sequence. [4]

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

The \(n\)th term of a sequence, \(u_n\), is given by $$u_n = 12 - \frac{1}{2}n.$$

1. Write down the values of \(u_1\), \(u_2\) and \(u_3\). State what type of sequence this is. [2]
2. Find \(\sum_{n=1}^{30} u_n\). [3]

An arithmetic progression (AP) and a geometric progression (GP) have the same first and fourth terms as each other. The first term of both is 1.5 and the fourth term of both is 12. Calculate the difference between the tenth terms of the AP and the GP. [5]