1.04h Arithmetic sequences: nth term and sum formulae

8 The first term of a progression is \(4 x\) and the second term is \(x ^ { 2 }\).

1. For the case where the progression is arithmetic with a common difference of 12 , find the possible values of \(x\) and the corresponding values of the third term.
2. For the case where the progression is geometric with a sum to infinity of 8 , find the third term.

11. The first three terms of an arithmetic series are \(60,4 p\) and \(2 p - 6\) respectively.

1. Show that \(p = 9\)
2. Find the value of the 20th term of this series.
3. Prove that the sum of the first \(n\) terms of this series is given by the expression $$12 n ( 6 - n )$$ \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-27_106_68_2615_1877}

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

4. The \(4 ^ { \text {th } }\) term of an arithmetic sequence is 3 and the sum of the first 6 terms is 27 Find the first term and the common difference of this sequence.

4. An arithmetic series has first term \(a\) and common difference \(d\). Given that the sum of the first 9 terms is 54

1. show that $$a + 4 d = 6$$ Given also that the 8th term is half the 7th term,
2. find the values of \(a\) and \(d\).

12. Karen is going to raise money for a charity. She aims to cycle a total distance of 1000 km over a number of days.
On day one she cycles 25 km .
She increases the distance that she cycles each day by \(10 \%\) of the distance cycled on the previous day, until she reaches the total distance of 1000 km . She reaches the total distance of 1000 km on day \(N\), where \(N\) is a positive integer.

1. Find the value of \(N\). On day one, 50 people donated money to the charity. Each day, 20 more people donated to the charity than did so on the previous day, so that 70 people donated money on day two, 90 people donated money on day three, and so on.
2. Find the number of people who donated to the charity on day fifteen. Each day, the donation given by each person was \(\pounds 5\)
3. Find the total amount of money donated by the end of day fifteen.

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 \(a\).

8. A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find

1. the number of houses built in 1986, the first year of the building programme,
2. the total number of houses built in the 25 years of the programme.

14. The 5 th term of an arithmetic series is \(4 k\), where \(k\) is a constant. The sum of the first 8 terms of this series is \(20 k + 16\)

1. 1. Find, in terms of \(k\), an expression for the common difference of the series.
   2. Show that the first term of the series is \(16 - 8 k\) Given that the 9th term of the series is 24, find
2. the value of \(k\),
3. the sum of the first 20 terms. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-40_2257_54_314_1977}

1. In a large theatre there are 20 rows of seats.

The number of seats in the first row is \(a\), where \(a\) is a constant. In the second row the number of seats is \(( a + d )\), where \(d\) is a constant. In the third row the number of seats is \(( a + 2 d )\), and on each subsequent row there are \(d\) more seats than on the previous row. The number of seats in each row forms an arithmetic sequence. The total number of seats in the first 10 rows is 395

1. Use this information to show that \(10 a + 45 d = 395\) The total number of seats in the first 18 rows is 927
2. Use this information to write down a second simplified equation relating \(a\) and \(d\).
3. Solve these equations to find the value of \(a\) and the value of \(d\).
4. Find the number of seats in the 20th row of the theatre.

9. A car manufacturer currently makes 1000 cars each week. The manufacturer plans to increase the number of cars it makes each week. The number of cars made will be increased by 20 each week from 1000 in week 1, to 1020 in week 2, to 1040 in week 3 and so on, until 1500 cars are made in week \(N\).

1. Find the value of \(N\). The car manufacturer then plans to continue to make 1500 cars each week.
2. Find the total number of cars that will be made in the first 50 weeks starting from and including week 1.

16. Maria trains for a triathlon, which involves swimming, cycling and running. On the first day of training she swims 1.5 km and then she swims 1.5 km on each of the following days.

1. Find the total distance that Maria swims in the first 17 days of training. Maria also runs 1.5 km on the first day of training and on each of the following days she runs 0.25 km further than on the previous day. So she runs 1.75 km on the second day and 2 km on the third day and so on.
2. Find how far Maria runs on the 17th day of training. Maria also cycles 1.5 km on the first day of training and on each of the following days she cycles \(5 \%\) further than on the previous day.
3. Find the total distance that Maria cycles in the first 17 days of training.
4. Find the total distance Maria travels by swimming, running and cycling in the first 17 days of training. Maria needs to cycle 40 km in the triathlon.
5. On which day of training does Maria first cycle more than 40 km ?

5. The \(r\) th term of an arithmetic series is ( \(2 r - 5\) ).

1. Write down the first three terms of this series.
2. State the value of the common difference.
3. Show that \(\sum _ { r = 1 } ^ { n } ( 2 r - 5 ) = n ( n - 4 )\).

9. Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns: Row 1 □ Row 2 □ 1 Row 3 \includegraphics[max width=\textwidth, alt={}, center]{fff086fd-f5d8-45b7-8db1-8b22ba5aab31-11_40_104_566_479} She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make 2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.

1. Find an expression, in terms of \(n\), for the number of sticks required to make a similar arrangement of \(n\) squares in the \(n\)th row. Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.
2. Find the total number of sticks Ann uses in making these 10 rows. Ann started with 1750 sticks. Given that Ann continues the pattern to complete \(k\) rows but does not have sufficient sticks to complete the ( \(k + 1\) )th row,
3. show that \(k\) satisfies \(( 3 k - 100 ) ( k + 35 ) < 0\).
4. Find the value of \(k\).

11. The first term of an arithmetic sequence is 30 and the common difference is - 1.5

1. Find the value of the 25th term. The \(r\) th term of the sequence is 0 .
2. Find the value of \(r\). The sum of the first \(n\) terms of the sequence is \(S _ { n }\).
3. Find the largest positive value of \(S _ { n }\).

9. The first term of an arithmetic series is \(a\) and the common difference is \(d\). The 18th term of the series is 25 and the 21st term of the series is \(32 \frac { 1 } { 2 }\).

1. Use this information to write down two equations for \(a\) and \(d\).
2. Show that \(a = - 17.5\) and find the value of \(d\). The sum of the first \(n\) terms of the series is 2750 .
3. Show that \(n\) is given by $$n ^ { 2 } - 15 n = 55 \times 40 .$$
4. Hence find the value of \(n\).

7. Jill gave money to a charity over a 20 -year period, from Year 1 to Year 20 inclusive. She gave \(\pounds 150\) in Year \(1 , \pounds 160\) in Year 2, \(\pounds 170\) in Year 3, and so on, so that the amounts of money she gave each year formed an arithmetic sequence.

1. Find the amount of money she gave in Year 10.
2. Calculate the total amount of money she gave over the 20 -year period. Kevin also gave money to the charity over the same 20 -year period. He gave \(\pounds A\) in Year 1 and the amounts of money he gave each year increased, forming an arithmetic sequence with common difference \(\pounds 30\). The total amount of money that Kevin gave over the 20 -year period was twice the total amount of money that Jill gave.
3. Calculate the value of \(A\).

6. An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 10 terms of the sequence is 162 .

1. Show that \(10 a + 45 d = 162\) Given also that the sixth term of the sequence is 17 ,
2. write down a second equation in \(a\) and \(d\),
3. find the value of \(a\) and the value of \(d\).

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 \(P\).

1. Lewis played a game of space invaders. He scored points for each spaceship that he captured.

Lewis scored 140 points for capturing his first spaceship.
He scored 160 points for capturing his second spaceship, 180 points for capturing his third spaceship, and so on. The number of points scored for capturing each successive spaceship formed an arithmetic sequence.

1. Find the number of points that Lewis scored for capturing his 20th spaceship.
2. Find the total number of points Lewis scored for capturing his first 20 spaceships. Sian played an adventure game. She scored points for each dragon that she captured. The number of points that Sian scored for capturing each successive dragon formed an arithmetic sequence. Sian captured \(n\) dragons and the total number of points that she scored for capturing all \(n\) dragons was 8500 . Given that Sian scored 300 points for capturing her first dragon and then 700 points for capturing her \(n\)th dragon,
3. find the value of \(n\).

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.