Arithmetic Sequences and Series

2. A sequence is defined by

6 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.

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 \(\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }\)

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. 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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3 Ara rith etic p og essich s first term 7 Th \(n\)th erm is \& d (3n)ttl erm is \% Fid b lue \(6 n\).

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.

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

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

4. (a) Show that \(\sum _ { r = 1 } ^ { 20 } \left( 2 ^ { r - 1 } - 3 - 4 r \right) = 1047675\) (b) A sequence has \(n\)th term \(u _ { n } = \sin \left( 90 n ^ { \circ } \right) n \geq 1\)

1. Find the order of the sequence.
2. Find \(\sum _ { r = 1 } ^ { 222 } u _ { r }\)

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. The second, third and fourth terms of an arithmetic sequence are \(2 k , 5 k - 10\) and \(7 k - 14\) respectively, where \(k\) is a constant.

Show that the sum of the first \(n\) terms of the sequence is a square number.

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).

5. Use the method of differences to prove that for \(n > 2\) $$\sum _ { r = 2 } ^ { n } \frac { 4 } { r ^ { 2 } - 1 } = \frac { ( p n + q ) ( n - 1 ) } { n ( n + 1 ) }$$ where \(p\) and \(q\) are constants to be determined.
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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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5. Use the method of differences to prove that for \(n > 2\) $$\sum _ { r = 2 } ^ { n } \frac { 4 } { r ^ { 2 } - 1 } = \frac { ( p n + q ) ( n - 1 ) } { n ( n + 1 ) }$$ where \(p\) and \(q\) are constants to be determined.
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4 The positive integer \(k\) is such that $$\sum _ { r = 1 } ^ { k } ( 3 r - k ) = 90$$ Find the value of \(k\).

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 \(n\).

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

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.

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.

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.

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 \(\sum _ { i = 1 } ^ { 99 } a _ { i }\).

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

6 A sequence is given by $$\begin{gathered} a _ { 1 } = 4 \\ a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.

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.

4. Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045\).
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3 The first term of an arithmetic series is 1 . The common difference of the series is 6 .

1. Find the tenth term of the series.
2. The sum of the first \(n\) terms of the series is 7400 .
   1. Show that \(3 n ^ { 2 } - 2 n - 7400 = 0\).
   2. Find the value of \(n\).

1. Show that the common difference is 5 .
2. Find the 12th term. \end{enumerate} 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．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 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.

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.

13. (a) Find the sum of all the integers between 1 and 1000 which are divisible by 7 .
(b) Hence, or otherwise, evaluate \(\sum _ { r = 1 } ^ { 142 } ( 7 r + 2 )\).
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14

1. Use the laws of logarithms to show that \(\log _ { 10 } 200 - \log _ { 10 } 20\) is equal to 1 . The first three terms of a sequence are \(\log _ { 10 } 20 , \log _ { 10 } 200 , \log _ { 10 } 2000\).
2. Show that the sequence is arithmetic.
3. Find the exact value of the sum of the first 50 terms of this sequence.