1.04h Arithmetic sequences: nth term and sum formulae

6. (a) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]\). A company made a profit of \(\pounds 54000\) in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference \(\pounds d\). This model predicts total profits of \(\pounds 619200\) for the 9 years 2001 to 2009 inclusive.
(b) Find the value of \(d\). Using your value of \(d\),
(c) find the predicted profit for the year 2011.

9. (a) Prove that the sum of the first \(n\) terms of an arithmetic series with first term \(a\) and common difference \(d\) is given by $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ A novelist begins writing a new book. She plans to write 16 pages during the first week, 18 during the second and so on, with the number of pages increasing by 2 each week. Find, according to her plan,
(b) how many pages she will write in the fifth week,
(c) the total number of pages she will write in the first five weeks.
(d) Using algebra, find how long it will take her to write the book if it has 250 pages.

7. (a) An arithmetic series has a common difference of 7 . Given that the sum of the first 20 terms of the series is 530 , find

1. the first term of the series,
2. the smallest positive term of the series.
   (b) The terms of a sequence are given by $$u _ { n } = ( n + k ) ^ { 2 } , \quad n \geq 1 ,$$ where \(k\) is a positive constant.
   Given that \(u _ { 2 } = 2 u _ { 1 }\),
3. find the value of \(k\),
4. show that \(u _ { 3 } = 11 + 6 \sqrt { 2 }\).

8. (a) The first and third terms of an arithmetic series are 3 and 27 respectively.

1. Find the common difference of the series.
2. Find the sum of the first 11 terms of the series.
   (b) Find the sum of the integers between 50 and 150 which are divisible by 8 .

7. As part of a new training programme, Habib decides to do sit-ups every day. He plans to do 20 per day in the first week, 22 per day in the second week, 24 per day in the third week and so on, increasing the daily number of sit-ups by two at the start of each week.

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 \(n\).

3 An arithmetic series has fifth term 46 and twentieth term 181.

1. 1. Show that the common difference is 9 .
   2. Find the first term.
2. Find the sum of the first 20 terms of the series.
3. The \(n\)th term of the series is \(u _ { n }\). Given that the sum of the first 50 terms of the series is 11525 , find the value of $$\sum _ { n = 21 } ^ { 50 } u _ { n }$$

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.

8 The 25th term of an arithmetic series is 38 .
The sum of the first 40 terms of the series is 1250 .

1. Show that the common difference of this series is 1.5 .
2. Find the number of terms in the series which are less than 100 .

4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 31 terms of the series is 310 .

1. Show that \(a + 15 d = 10\).
2. Given also that the 21st term is twice the 16th term, find the value of \(d\).
3. The \(n\)th term of the series is \(u _ { n }\). Given that \(\sum _ { n = 1 } ^ { k } u _ { n } = 0\), find the value of \(k\).

6 An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 25 terms of the series is 3500 .

1. Show that \(a + 12 d = 140\).
2. The fifth term of this series is 100 . Find the value of \(d\) and the value of \(a\).
3. The \(n\)th term of this series is \(u _ { n }\). Given that $$33 \left( \sum _ { n = 1 } ^ { 25 } u _ { n } - \sum _ { n = 1 } ^ { k } u _ { n } \right) = 67 \sum _ { n = 1 } ^ { k } u _ { n }$$ find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
   (3 marks)

6

1. A geometric series begins \(420 + 294 + 205.8 + \ldots\).
   1. Show that the common ratio of the series is 0.7 .
   2. Find the sum to infinity of the series.
   3. Write the \(n\)th term of the series in the form \(p \times q ^ { n }\), where \(p\) and \(q\) are constants.
2. The first term of an arithmetic series is 240 and the common difference of the series is - 8 . The \(n\)th term of the series is \(u _ { n }\).
   1. Write down an expression for \(u _ { n }\).
   2. Given that \(u _ { k } = 0\), find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).

3 The \(n\)th term of an arithmetic sequence is \(u _ { n }\), where $$u _ { n } = 90 - 3 n$$

1. Find the value of \(u _ { 1 }\) and the value of \(u _ { 2 }\).
2. Write down the common difference of the arithmetic sequence.
3. Given that \(\sum _ { n = 1 } ^ { k } u _ { n } = 0\), find the value of \(k\).

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 \(n\).

5

1. An infinite geometric series has common ratio \(r\).
   The first term of the series is 10 and its sum to infinity is 50 .
   1. Show that \(r = \frac { 4 } { 5 }\).
   2. Find the second term of the series.
2. The first and second terms of the geometric series in part (a) have the same values as the 4th and 8th terms respectively of an arithmetic series.
   1. Find the common difference of the arithmetic series.
   2. The \(n\)th term of the arithmetic series is \(u _ { n }\). Find the value of \(\sum _ { n = 1 } ^ { 40 } u _ { n }\).

1 The arithmetic series $$23 + 32 + 41 + 50 + \ldots + 2534$$ has 280 terms.

1. Write down the common difference of the series.
2. Find the 100th term of the series.
3. Find the sum of the 280 terms of the series.

8 An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 5 terms of the series is 575 .

1. Show that \(a + 2 d = 115\).
2. Given also that the 10th term of the series is 87, find the value of \(d\).
3. The \(n\)th term of the series is \(u _ { n }\). Given that \(u _ { k } > 0\) and \(u _ { k + 1 } < 0\), find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
   [0pt] [5 marks]

4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 21 terms is 168 .

1. Show that \(a + 10 d = 8\).
2. The sum of the second term and the third term is 50 . The \(n\)th term of the series is \(u _ { n }\).
   1. Find the value of \(u _ { 12 }\).
   2. Find the value of \(\sum _ { n = 4 } ^ { 21 } u _ { n }\).

8. (a) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]\). A company made a profit of \(\pounds 54000\) in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference \(\pounds d\). This model predicts total profits of \(\pounds 619200\) for the 9 years 2001 to 2009 inclusive.
(b) Find the value of \(d\). Using your value of \(d\),
(c) find the predicted profit for the year 2011. 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 \(\pounds 54000\),
(d) find the predicted profit for the year 2011.

1. An art display consists of an arrangement of \(n\) marbles.

When arranged in ascending order of mass, the mass of the first marble is 10 grams. The mass of each subsequent marble is 3 grams more than the mass of the previous one, so that the \(r\) th marble has mass \(( 7 + 3 r )\) grams.

1. Show that the mean mass, in grams, of the marbles in the display is given by $$\frac { 1 } { 2 } ( 3 n + 17 )$$ Given that there are 85 marbles in the display,
2. use the standard summation formulae to find the standard deviation of the mass of the marbles in the display, giving your answer, in grams, to one decimal place.

5 An ice cream seller expects that the number of sales will increase by the same amount every week from May onwards. 150 ice creams are sold in Week 1 and 166 ice creams are sold in Week 2. The ice cream seller makes a profit of \(\pounds 1.25\) for each ice cream sold.

1. Find the expected profit in Week 10.
2. In which week will the total expected profits first exceed \(\pounds 5000\) ?
3. Give two reasons why this model may not be appropriate.

11 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 \(g _ { 1 } = a _ { 1 } = 1 + \sqrt { 5 } , g _ { 3 } = a _ { 2 }\) and \(g _ { 4 } + a _ { 3 } = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2 \sqrt { 5 }\).