Prove sum formula

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

1. A company offers two salary schemes for a 10 -year period, Year 1 to Year 10 inclusive.

Scheme 1: Salary in Year 1 is \(\pounds P\).
Salary increases by \(\pounds ( 2 T )\) each year, forming an arithmetic sequence.
Scheme 2: Salary in Year 1 is \(\pounds ( P + 1800 )\).
Salary increases by \(\pounds T\) each year, forming an arithmetic sequence.

1. Show that the total earned under Salary Scheme 1 for the 10-year period is $$\pounds ( 10 P + 90 T )$$ For the 10-year period, the total earned is the same for both salary schemes.
2. Find the value of \(T\). For this value of \(T\), the salary in Year 10 under Salary Scheme 2 is \(\pounds 29850\)
3. Find the value of \(P\).

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.

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 \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .\) (4)
   A polygon has 16 sides. The lengths of the sides of the polygon, starting with the shortest side, form an arithmetic sequence with common difference \(d \mathrm {~cm}\).
   The longest side of the polygon has length 6 cm and the perimeter of the polygon is 72 cm .
   Find
2. the length of the shortest side of the polygon,
   (5)
3. the value of \(d\).
   (2) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \end{tabular} & Leave blank
   \hline \end{tabular} \end{center}

   Leave blank

   \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

8. (i) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum to \(n\) terms of this series is $$\frac { n } { 2 } \{ 2 a + ( n - 1 ) d \}$$ (ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by $$u _ { n } = 5 n + 3 ( - 1 ) ^ { n }$$ Find the value of

1. \(u _ { 5 }\)
2. \(\sum _ { n = 1 } ^ { 59 } u _ { n }\)

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.
   
   \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-16_2673_1948_107_118}

1. 1. In an arithmetic series, the first term is \(a\) and the common difference is \(d\).

Show that $$S _ { n } = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$$ (ii) James saves money over a number of weeks to buy a printer that costs \(\pounds 64\) He saves \(\pounds 10\) in week \(1 , \pounds 9.20\) in week \(2 , \pounds 8.40\) in week 3 and so on, so that the weekly amounts he saves form an arithmetic sequence. Given that James takes \(n\) weeks to save exactly \(\pounds 64\)

1. show that $$n ^ { 2 } - 26 n + 160 = 0$$
2. Solve the equation $$n ^ { 2 } - 26 n + 160 = 0$$
3. Hence state the number of weeks James takes to save enough money to buy the printer, giving a brief reason for your answer.

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.

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.

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

1. Show that $$( r + 1 ) ^ { 2 } - r ^ { 2 } = 2 r + 1$$ 4
2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } ( 2 r + 1 ) = n ^ { 2 } + 2 n$$ 4
3. Verify that using the formula for \(\sum _ { r = 1 } ^ { n } r\) gives the same result as that given in part (b).
   [0pt] [3 marks]

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 $$\frac{1}{2}n[2a + (n - 1)d].$$ [4]

Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays £149 in the first month, £147 in the second month, £145 in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).

1. Find the amount Sean repays in the 21st month. [2]

Over the \(n\) months, he repays a total of £5000.

1. Form an equation in \(n\), and show that your equation may be written as $$n^2 - 150n + 5000 = 0.$$ [3]
2. Solve the equation in part (c). [3]
3. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem. [1]

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]

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]