1.04h Arithmetic sequences: nth term and sum formulae

2

1. Ben saves his pocket money as follows.
   Each week he puts money into his piggy bank (which pays no interest). In the first week he puts in 10p. In the second week he puts in 12p. In the third week he puts in 14p, and so on. How much money does Ben have in his piggy bank after 25 weeks?
2. On January 1st Shirley invests \(\pounds 500\) in a savings account that pays compound interest at \(3 \%\) per annum. She makes no further payments into this account. The interest is added on 31st December each year.
   1. Find the number of years after which her investment will first be worth more than \(\pounds 600\).
   2. State an assumption that you have made in answering part (ii)(a).

9. An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays \(\pounds 149\) in the first month, \(\pounds 147\) in the second month, \(\pounds 145\) in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).
2. Find the amount Sean repays in the 21st month. Over the \(n\) months, he repays a total of \(\pounds 5000\).
3. Form an equation in \(n\), and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0$$
4. Solve the equation in part (c).
5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.

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

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

7 Business A made a \(\pounds 5000\) profit during its first year.
In each subsequent year, the profit increased by \(\pounds 1500\) so that the profit was \(\pounds 6500\) during the second year, \(\pounds 8000\) during the third year and so on. Business B made a \(\pounds 5000\) 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. Find an expression for the total profit made by business B during the first \(n\) years. Give your answer in its simplest form.
3. Find how many years it will take for the total profit of business A to reach \(\pounds 385000\).
4. Comment on the profits made by each business in the long term.

9 An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 36 terms of the sequence is equal to the square of the sum of the first 6 terms. 9

1. Show that \(4 a + 70 d = 4 a ^ { 2 } + 20 a d + 25 d ^ { 2 }\) 9
2. Given that the sixth term of the sequence is 25 , find the smallest possible value of \(a\).

10

1. An arithmetic series is given by $$\sum _ { r = 5 } ^ { 20 } ( 4 r + 1 )$$ 10
   1. (i) Write down the first term of the series.
      10
   2. (ii) Write down the common difference of the series.
      10
   3. (iii) Find the number of terms of the series.
      

      10

      A different arithmetic series is given by \(\sum _ { r = 10 } ^ { 100 } ( b r + c )\) where \(b\) and \(c\) are constants. The sum of this series is 7735

      10
   4. (ii) The 40th term of the series is 4 times the 2nd term. Find the values of \(b\) and \(c\).
      [0pt] [4 marks]

6

1. The ninth term of an arithmetic series is 3 The sum of the first \(n\) terms of the series is \(S _ { n }\) and \(S _ { 21 } = 42\) Find the first term and common difference of the series.
   [0pt] [4 marks]
   6
2. A second arithmetic series has first term - 18 and common difference \(\frac { 3 } { 4 }\) The sum of the first \(n\) terms of this series is \(T _ { n }\) Find the value of \(n\) such that \(T _ { n } = S _ { n }\) [0pt] [3 marks]

9 The first three terms of an arithmetic sequence are given by $$2 x + 5 \quad 5 x + 1 \quad 6 x + 7$$ 9

1. Show that \(x = 5\) is the only value which gives an arithmetic sequence.
   9
2. (i) Write down the value of the first term of the sequence.
   9 (b) (ii) Find the value of the common difference of the sequence.
   9
3. The sum of the first \(N\) terms of the arithmetic sequence is \(S _ { N }\) where $$\begin{array} { r } S _ { N } < 100000 \\ S _ { N + 1 } > 100000 \end{array}$$ Find the value of \(N\).
   [0pt] [4 marks]

5 Ziad is training to become a long-distance swimmer. He trains every day by swimming lengths at his local pool.
The length of the pool is 25 metres.
Each day he increases the number of lengths that he swims by four.
On his first day of training, Ziad swims 10 lengths of the pool.
5

1. Write down an expression for the number of lengths Ziad will swim on his \(n\)th day of training. 5
2. (i) Ziad's target is to be able to swim at least 3000 metres in one day.
   Determine the minimum number of days he will need to train to reach his target.
   5 (b) (ii) Ziad's coach claims that when he reaches his target he will have covered a total distance of over 50000 metres. Determine if Ziad's coach is correct.

1. Jem pays money into a savings scheme, \(A\), over a period of 300 months.

Jem pays \(\pounds 20\) into scheme \(A\) in month \(1 , \pounds 20.50\) in month \(2 , \pounds 21\) in month 3 and so on, so that the amounts Jem pays each month form an arithmetic sequence.

1. Show that Jem pays \(\pounds 69.50\) into scheme \(A\) in month 100
2. Find the total amount that Jem pays into scheme \(A\) over the period of 300 months. Kim pays money into a different savings scheme, \(B\), over the same period of 300 months. In a model, the amounts Kim pays into scheme \(B\) increase by the same percentage each month, so that the amounts Kim pays each month form a geometric sequence. Given that Kim pays
   • \(\pounds 20\) into scheme \(B\) in month 1
   • \(\pounds 250\) into scheme \(B\) in month 300
   • use the model to calculate, to the nearest \(\pounds 10\), the difference between the total amount paid into scheme \(A\) and the total amount paid into scheme \(B\) over the period of 300 months.

3

1. In an arithmetic progression, the first term is 7 and the sum of the first 40 terms is 4960. Find the common difference.
2. A geometric progression has first term 14 and common ratio 0.3. Find the sum to infinity.

2

1. An arithmetic sequence has first term 3 and common difference 2. Find the twenty-first term of this sequence.
2. Find the sum to infinity of a geometric progression with first term 162 and second term 54.
3. A sequence is given by the recurrence relation \(u _ { 1 } = 3 , u _ { n + 1 } = 2 - u _ { n } , n = 1,2,3 , \ldots\). Find \(u _ { 2 } , u _ { 3 }\), \(u _ { 4 } , u _ { 5 }\) and describe the behaviour of this sequence.

5 An arithmetic progression has first term 5 and common difference 7.

1. Find the value of the 10th term.
2. Find the sum of the first 15 terms. The terms of the progression are given by \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\).
3. Evaluate \(\sum _ { n = 1 } ^ { 15 } \left( 2 x _ { n } + 1 \right)\).

4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), is defined by \(u _ { n } = 3 n + 5\).

1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find the value of \(n\) such that \(u _ { n } = 254\).
3. Evaluate \(\sum _ { n = 1 } ^ { 500 } u _ { n }\).

11 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 \(r = \frac { 5 } { 8 }\).
2. Find the sum to infinity of the geometric progression in terms of \(a\).

10 An arithmetic sequence and a geometric sequence have \(n\)th terms \(a _ { n }\) and \(g _ { n }\) respectively, where \(n = 1,2,3 , \ldots\). It is given that \(a _ { 1 } = g _ { 1 } , a _ { 2 } = g _ { 2 } , a _ { 5 } = g _ { 3 } , a _ { 1 } \neq a _ { 2 }\) and \(a _ { 1 } \neq 0\).

1. Show that the common ratio of the geometric sequence is 3 .
2. Find the common difference of the arithmetic sequence in terms of \(a _ { 1 }\).
3. Let \(a _ { 1 } = g _ { 1 } = 5\).
   1. Find the first three terms of both sequences.
   2. Show that every term of the geometric sequence is also a term of the arithmetic sequence.

11 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 \(r = \frac { 5 } { 8 }\).
2. Find the sum to infinity of the geometric progression in terms of \(a\).

6

1. Given that the numbers \(a , b\) and \(c\) are in arithmetic progression, show that \(a + c = 2 b\).
2. Find an analogous result for three numbers in geometric progression.
3. The numbers \(2 - 3 x , 2 x , 3 - 2 x\) are the first three terms of a convergent geometric progression. Find \(x\) and hence calculate the sum to infinity.

8

1. The sum of the first \(n\) terms of the arithmetic series \(1 + 3 + 5 + \ldots\) exceeds the sum of the first \(n\) terms of the arithmetic series \(100 + 97 + 94 + \ldots\). Find the least possible value of \(n\).
2. \(3 \sqrt { 2 }\) and \(2 - \sqrt { 2 }\) are the first two terms of a geometric progression.
   1. Show that the third term is \(\sqrt { 2 } - \frac { 4 } { 3 }\).
   2. Find the index \(n\) of the first term that is less than 0.01.
   3. Find the exact value of the sum to infinity of this progression.
   4. Which of the terms 'alternating', 'periodic', 'convergent' apply to the sequences generated by the following \(n\)th terms, where \(n\) is a positive integer?
      (a) \(1 - \left( \frac { 3 } { 4 } \right) ^ { n }\) (b) \(\frac { 1 } { n } \cos n \pi\) (c) \(\sec n \pi\)

a) The \(3 ^ { \text {rd } } , 19 ^ { \text {th } }\) and \(67 ^ { \text {th } }\) terms of an arithmetic sequence form a geometric sequence. Given that the arithmetic sequence is increasing and that the first term is 3 , find the common difference of the arithmetic sequence. b) A firm has 100 employees on a particular Monday. The next day it adds 12 employees onto its staff and continues to do so on every successive working day, from Monday to Friday.
i) Find the number of employees at the end of the \(8 { } ^ { \text {th } }\) week.
ii) Each employee is paid \(\pounds 55\) per working day. Determine the total wage bill for the 8 week period.

The first and second terms of an arithmetic progression are \(\tan\theta\) and \(\sin\theta\) respectively, where \(0 < \theta < \frac{1}{2}\pi\). \begin{enumerate}[label=(\alph*)] \item Given that \(\theta = \frac{1}{4}\pi\), find the exact sum of the first 40 terms of the progression. [4] \end enumerate} The first and second terms of a geometric progression are \(\tan\theta\) and \(\sin\theta\) respectively, where \(0 < \theta < \frac{1}{2}\pi\).

1. 1. Find the sum to infinity of the progression in terms of \(\theta\). [2]
   2. Given that \(\theta = \frac{1}{3}\pi\), find the sum of the first 10 terms of the progression. Give your answer correct to 3 significant figures. [3]

An arithmetic progression has first term 5 and common difference \(d\), where \(d > 0\). The second, fifth and eleventh terms of the arithmetic progression, in that order, are the first three terms of a geometric progression.

1. Find the value of \(d\). [3]
2. The sum of the first 77 terms of the arithmetic progression is denoted by \(S_{77}\). The sum of the first 10 terms of the geometric progression is denoted by \(G_{10}\). Find the value of \(S_{77} - G_{10}\). [5]

The first term of an arithmetic progression is \(-20\) and the common difference is \(5\).

1. Find the sum of the first 20 terms of the progression. [2]

It is given that the sum of the first \(2k\) terms is 10 times the sum of the first \(k\) terms.

1. Find the value of \(k\). [3]

An arithmetic progression has fourth term 15 and eighth term 25. Find the 30th term of the progression. [3]