Arithmetic Sequences and Series

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

1. An arithmetic sequence has first term 6 and common difference 10 Find
   1. the 15th term of the sequence,
   2. the sum of the first 20 terms of the sequence.
      

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 On his \(1 ^ { \text {st } }\) birthday, John was given \(\pounds 5\) by his Uncle Fred. On each succeeding birthday, Uncle Fred gave a sum of money that was \(\pounds 3\) more than the amount he gave on the last birthday.

1. How much did Uncle Fred give John on his \(8 { } ^ { \text {th } }\) birthday?
2. On what birthday did the gift from Uncle Fred result in the total sum given on all birthdays exceeding £200?

1. On Alice's 11th birthday she started to receive an annual allowance. The first annual allowance was \(\pounds 500\) and on each following birthday the allowance was increased by \(\pounds 200\).
   1. Show that, immediately after her 12th birthday, the total of the allowances that Alice had received was \(\pounds 1200\).
   2. Find the amount of Alice's annual allowance on her 18th birthday.
   3. Find the total of the allowances that Alice had received up to and including her 18th birthday.
   When the total of the allowances that Alice had received reached \(\pounds 32000\) the allowance stopped.
2. Find how old Alice was when she received her last allowance.

10. In this question you must show detailed reasoning. Owen wants to train for 12 weeks in preparation for running a marathon. During the 12-week period he will run every Sunday and every Wednesday.

• On Sunday in week 1 he will run 15 km
• On Sunday in week 12 he will run 37 km

He considers two different 12-week training plans. In training plan \(A\), he will increase the distance he runs each Sunday by the same amount.

1. Calculate the distance he will run on Sunday in week 5 under training plan \(A\). In training plan \(B\), he will increase the distance he runs each Sunday by the same percentage.
2. Calculate the distance he will run on Sunday in week 5 under training plan \(B\). Give your answer in km to one decimal place. Owen will also run a fixed distance, \(x \mathrm {~km}\), each Wednesday over the 12-week period. Given that
   • \(x\) is an integer
   • the total distance that Owen will run on Sundays and Wednesdays over the 12 weeks will not exceed 360 km
   • 1. find the maximum value of \(x\), if he uses training plan \(A\),
     2. find the maximum value of \(x\), if he uses training plan \(B\).
        
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5 A sequence is defined by \(a _ { k } = 5 k + 1\), for \(k = 1,2,3 \ldots\)

1. Write down the first three terms of the sequence.
2. Evaluate \(\sum _ { k = 1 } ^ { 100 } a _ { k }\).

6 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 2 n + 5 , \quad \text { for } n \geqslant 1 .$$

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. State what type of sequence it is.
3. Given that \(\sum _ { n = 1 } ^ { N } u _ { n } = 2200\), find the value of \(N\).

6 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 85 - 5 n\) for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
3. Given that \(u _ { 1 } , u _ { 5 }\) and \(u _ { p }\) are, respectively, the first, second and third terms of a geometric progression, find the value of \(p\).
4. Find the sum to infinity of the geometric progression in part (iii).

A sequence of terms \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 4$$ $$a_{n+1} = ka_n + 3$$ where \(k\) is a constant. Given that • \(\sum_{n=1}^{5} a_n = 12\) • all terms of the sequence are different find the value of \(k\) [4]

A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$a _ { 1 } = k , \quad a _ { n + 1 } = 4 a _ { n } - 7 ,$$ where \(k\) is a constant.

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Find \(a _ { 3 }\) in terms of \(k\), simplifying your answer. Given that \(a _ { 3 } = 13\),
3. find the value of \(k\).

A sequence of terms \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 4$$ $$a_{n+1} = ka_n + 3$$ where \(k\) is a constant. Given that • \(\sum_{n=1}^{5} a_n = 12\) • all terms of the sequence are different find the value of \(k\) [4]

2 The first, second and third terms of an arithmetic progression are \(a , 2 a\) and \(a ^ { 2 }\) respectively, where \(a\) is a positive constant. Find the sum of the first 50 terms of the progression.

4 The first, second and third terms of an arithmetic progression are \(k , 6 k\) and \(k + 6\) respectively.

1. Find the value of the constant \(k\).
2. Find the sum of the first 30 terms of the progression.

1. Three consecutive terms in an arithmetic sequence are \(3e^{-q}\), \(5\), \(3e^q\) Find the possible values of \(p\). Give your answers in an exact form. [6 marks]
2. Prove that there is no possible value of \(q\) for which \(3e^{-q}\), \(5\), \(3e^q\) are consecutive terms of a geometric sequence. [4 marks]

4 A progression has a first term of 12 and a fifth term of 18.

1. Find the sum of the first 25 terms if the progression is arithmetic.
2. Find the 13th term if the progression is geometric.

The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is

1. an arithmetic progression, [2]
2. a geometric progression. [2]

11 The curve \(y = \mathrm { f } ( x )\) is defined by the function \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\) with domain \(0 \leq x \leq 4 \pi\).

1. 1. Show that the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\), when arranged in increasing order, form an arithmetic sequence.
   2. Show that the corresponding \(y\)-coordinates form a geometric sequence.
2. Would the result still hold with a larger domain? Give reasons for your answer.

1 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).

2 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).

5. Given that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } a _ { r } = 12 + 4 n ^ { 2 }$$

1. find the value of \(\sum _ { r = 1 } ^ { 5 } a _ { r }\)
2. Find the value of \(a _ { 6 }\)

1. Find the number of sit-ups that Habib will do in the fifth week.
2. Show that he will do a total of 1512 sit-ups during the first eight weeks. In the \(n\)th week of training, the number of sit-ups that Habib does is greater than 300 for the first time.
3. Find the value of \(n\).

1. Each year, Andy pays into a savings scheme. In year one he pays in \(\pounds 600\). His payments increase by \(\pounds 120\) each year so that he pays \(\pounds 720\) in year two, \(\pounds 840\) in year three and so on, so that his payments form an arithmetic sequence.
   1. Find out how much Andy pays into the savings scheme in year ten.
      (2)
   Kim starts paying money into a different savings scheme at the same time as Andy. In year one she pays in \(\pounds 130\). Her payments increase each year so that she pays \(\pounds 210\) in year two, \(\pounds 290\) in year three and so on, so that her payments form a different arithmetic sequence. At the end of year \(N\), Andy has paid, in total, twice as much money into his savings scheme as Kim has paid, in total, into her savings scheme.
2. Find the value of \(N\).

9 A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.

1. On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day.
   1. How many litres will be lost on the 30th day after filling?
   2. The tank becomes empty during the \(n\)th day after filling. Find the value of \(n\).
   3. Assume instead that 10 litres of water are lost on the first day and that the amount of water lost increases by \(10 \%\) on each succeeding day. Find what percentage of the original 2000 litres is left in the tank at the end of the 30th day after filling.
      [0pt] [Questions 10 and 11 are printed on the next page.]

2. A sequence is defined by

Find the second and third terms in the sequence given by $$u_1 = 5,$$ $$u_{n+1} = u_n + 3.$$ Find also the sum of the first 50 terms of this sequence. [4]

The sequence \(u_1, u_2, u_3, ...\) is defined by the recurrence relation $$u_{n+1} = (u_n)^2 - 1, \quad n \geq 1.$$ Given that \(u_1 = k\), where \(k\) is a constant,

1. find expressions for \(u_2\) and \(u_3\) in terms of \(k\). [3]

Given also that \(u_2 + u_3 = 11\),

1. find the possible values of \(k\). [4]

1 A sequence is defined by \(u _ { 1 } = 2\) and \(u _ { k + 1 } = \frac { 10 } { u _ { k } ^ { 2 } }\).
Calculate \(\sum _ { k = 1 } ^ { 4 } u _ { k }\).

7. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3 \\ u _ { n + 1 } & = u _ { n } - 5 , \quad n \geqslant 1 \end{aligned}$$ Find the values of

1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. \(u _ { 100 }\)
3. \(\sum _ { i = 1 } ^ { 100 } u _ { i }\)

1. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by

$$\begin{aligned} a _ { 1 } & = 3 \\ a _ { n + 1 } & = \frac { a _ { n } - 3 } { a _ { n } - 2 } , \quad n \in \mathbb { N } \end{aligned}$$

1. Find \(\sum _ { r = 1 } ^ { 100 } a _ { r }\)
2. Hence find \(\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }\)

5. (a) Prove that the sum of the first \(n\) terms of an arithmetic series is given by the formula $$S _ { n } = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$$ where \(a\) is the first term of the series and \(d\) is the common difference between the terms.
(b) Find the sum of the integers which are divisible by 7 and lie between 1 and 500

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 $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$$ 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 \(N\).
2. Find the value of \(N\) The company then plans to continue to make 600 mobile phones each week.
3. Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.
   
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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]

2 A sequence begins $$\begin{array} { l l l l l l l l l l l l } 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 & 1 & \ldots \end{array}$$ and continues in this pattern.

1. Find the 48th term of this sequence.
2. Find the sum of the first 48 terms of this sequence.

2 A sequence begins $$\begin{array} { l l l l l l l l l l l l } 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 & 1 & \ldots \end{array}$$ and continues in this pattern.

1. Find the 48th term of this sequence.
2. Find the sum of the first 48 terms of this sequence.

11 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \ldots\) where \(a _ { 1 }\) is a given real number is defined by \(a _ { n + 1 } = 1 - \frac { 1 } { a _ { n } }\).

1. For the case when \(a _ { 1 } = 2\), find \(a _ { 2 } , a _ { 3 }\) and \(a _ { 4 }\). Describe the behaviour of this sequence
2. For the case when \(a _ { 1 } = k\), where \(k\) is an integer greater than 1 , find \(a _ { 2 }\) in terms of \(k\) as a single fraction.
   Find also \(a _ { 3 }\) in its simplest form and hence deduce that \(a _ { 4 } = k\).
3. Show that \(a _ { 2 } a _ { 3 } a _ { 4 } = - 1\) for any integer \(k\).
4. When \(a _ { 1 } = 2\) evaluate \(\sum _ { i = 1 } ^ { 99 } a _ { i }\).

1 The first term of an arithmetic progression is 61 and the second term is 57. The sum of the first \(n\) terms in \(n\). Find the value of the positive integer \(n\).

3 The first term of an arithmetic progression is 6 and the fifth term is 12 . The progression has \(n\) terms and the sum of all the terms is 90 . Find the value of \(n\).

1 The first term of an arithmetic progression is 61 and the second term is 57. The sum of the first \(n\) terms in \(n\). Find the value of the positive integer \(n\).

2 The \(n\)th term of an arithmetic progression is \(6 + 5 n\). Find the sum of the first 20 terms.

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).

2 The thirteenth term of an arithmetic progression is 12 and the sum of the first 30 terms is - 15 .
Find the sum of the first 50 terms of the progression.

1. Show that the common difference is 5 .
2. Find the 12th term. Another arithmetic sequence has first term -12 and common difference 7 .
   Given that the sums of the first \(n\) terms of these two sequences are equal,
3. find the value of \(n\).

1. A farmer has a pay scheme to keep fruit pickers working throughout the 30 day season. He pays \(\pounds a\) for their first day, \(\pounds ( a + d )\) for their second day, \(\pounds ( a + 2 d )\) for their third day, and so on, thus increasing the daily payment by \(\pounds d\) for each extra day they work.

A picker who works for all 30 days will earn \(\pounds 40.75\) on the final day.

1. Use this information to form an equation in \(a\) and \(d\). A picker who works for all 30 days will earn a total of \(\pounds 1005\)
2. Show that \(15 ( a + 40.75 ) = 1005\)
3. Hence find the value of \(a\) and the value of \(d\).

2.The sum of the first \(p\) terms of an arithmetic series is \(q\) and the sum of the first \(q\) terms of the same arithmetic series is \(p\) ,where \(p\) and \(q\) are positive integers and \(p \neq q\) . Giving simplified answers in terms of \(p\) and \(q\) ,find

1. the common difference of the terms in this series,
2. the first term of the series,
3. the sum of the first \(( p + q )\) terms of the series.

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

2 The arithmetic series $$51 + 58 + 65 + 72 + \ldots + 1444$$ has 200 terms.

1. Write down the common difference of the series.
2. Find the 101st term of the series.
3. Find the sum of the last 100 terms of the series.

7 The first term of an arithmetic progression is 1.5 and the sum of the first ten terms is 127.5 .

1. Find the common difference.
2. Find the sum of all the terms of the arithmetic progression whose values are between 25 and 100 .

Calculate \(\sum_{r=1}^{20} 5 + 2r\) [3]

2 Find the second and third terms in the sequence given by $$\begin{aligned} & u _ { 1 } = 5 \\ & u _ { n + 1 } = u _ { n } + 3 . \end{aligned}$$ Find also the sum of the first 50 terms of this sequence.

9. (i) Find the value of \(\sum _ { r = 1 } ^ { 20 } ( 3 + 5 r )\) (ii) Given that \(\sum _ { r = 0 } ^ { \infty } \frac { a } { 4 ^ { r } } = 16\), find the value of the constant \(a\).

3 An arithmetic progression has first term 4 and common difference \(d\). The sum of the first \(n\) terms of the progression is 5863.

1. Show that \(( n - 1 ) d = \frac { 11726 } { n } - 8\).
2. Given that the \(n\)th term is 139 , find the values of \(n\) and \(d\), giving the value of \(d\) as a fraction.

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

1. Find out how much Abbie pays into the savings scheme in the tenth year. Abbie pays into the scheme for \(n\) years until she has paid in a total of \(\pounds 67200\).
2. Show that \(n ^ { 2 } + 4 n - 24 \times 28 = 0\)
3. Hence find the number of years that Abbie pays into the savings scheme.

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 \(a\).

$$P(n) = \sum_{k=0}^{n} k^3 - \sum_{k=0}^{n-1} k^3 \text{ where } n \text{ is a positive integer.}$$

1. Find P(3) and P(10) [2 marks]
2. Solve the equation \(P(n) = 1.25 \times 10^8\) [2 marks]

5 The first, third and fifth terms of an arithmetic progression are \(2 \cos x , - 6 \sqrt { 3 } \sin x\) and \(10 \cos x\) respectively, where \(\frac { 1 } { 2 } \pi < x < \pi\).

1. Find the exact value of \(x\).
2. Hence find the exact sum of the first 25 terms of the progression.

The first and second terms of an arithmetic series are 200 and 197.5 respectively. The sum to \(n\) terms of the series is \(S_n\). Find the largest positive value of \(S_n\). [5]

1. Find the sum of all the integers between 1 and 1000 which are divisible by 7. [3]
2. Hence, or otherwise, evaluate \(\sum_{r=1}^{142} (7r + 2)\). [3]

9. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
The first 3 terms of an arithmetic sequence are $$\ln 3 \quad \ln \left( 3 ^ { k } - 1 \right) \quad \ln \left( 3 ^ { k } + 5 \right)$$ Find the exact value of the constant \(k\).
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