1.04i Geometric sequences: nth term and finite series sum

4 The diagram shows the curve with equation \(\mathrm { y } = 2 ^ { \mathrm { x } }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\). \includegraphics[max width=\textwidth, alt={}, center]{69c540e1-1dad-45a1-9809-7629d16260e0-06_824_1161_376_450}

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } 2 ^ { x } d x < U _ { N }\), where $$\mathrm { U } _ { \mathrm { N } } = \frac { 2 ^ { \frac { 1 } { \mathrm {~N} } } } { \mathrm {~N} \left( 2 ^ { \frac { 1 } { \mathrm {~N} } } - 1 \right) }$$
2. Use a similar method to find, in terms of \(N\), a lower bound \(\mathrm { L } _ { \mathrm { N } }\) for \(\int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x\).
3. Find the least value of \(N\) such that \(\mathrm { U } _ { \mathrm { N } } - \mathrm { L } _ { \mathrm { N } } < 10 ^ { - 4 }\).

9. In the first month after opening, a mobile phone shop sold 300 phones. A model for future sales assumes that the number of phones sold will increase by \(5 \%\) per month, so that \(300 \times 1.05\) will be sold in the second month, \(300 \times 1.05 ^ { 2 }\) in the third month, and so on. Using this model, calculate

1. the number of phones sold in the 24th month,
2. the total number of phones sold over the whole 24 months. This model predicts that, in the \(N\) th month, the number of phones sold in that month exceeds 3000 for the first time.
3. Find the value of \(N\).

12. A business is expected to have a yearly profit of \(\pounds 275000\) for the year 2016. The profit is expected to increase by \(10 \%\) per year, so that the expected yearly profits form a geometric sequence with common ratio 1.1

1. Show that the difference between the expected profit for the year 2020 and the expected profit for the year 2021 is \(\pounds 40300\) to the nearest hundred pounds.
2. Find the first year for which the expected yearly profit is more than one million pounds.
3. Find the total expected profits for the years 2016 to 2026 inclusive, giving your answer to the nearest hundred pounds.

9. The resident population of a city is 130000 at the end of Year 1 A model predicts that the resident population of the city will increase by \(2 \%\) each year, with the populations at the end of each year forming a geometric sequence.

1. Show that the predicted resident population at the end of Year 2 is 132600
2. Write down the value of the common ratio of the geometric sequence. The model predicts that Year \(N\) will be the first year which will end with the resident population of the city exceeding 260000
3. Show that $$N > \frac { \log _ { 10 } 2 } { \log _ { 10 } 1.02 } + 1$$
4. Find the value of \(N\).
   

14. A geometric series has a first term \(a\) and a common ratio \(r\).

1. Prove that the sum of the first \(n\) terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ A liquid is to be stored in a barrel. Due to evaporation, the volume of the liquid in a barrel at the end of a year is \(7 \%\) less than the volume at the start of the year. At the start of the first year, a barrel is filled with 180 litres of the liquid.
2. Show that the amount of the liquid in this barrel at the end of 5 years is approximately 125.2 litres. At the start of each year a new identical barrel is filled with 180 litres of the liquid so that, at the end of 20 years, there are 20 barrels containing varying amounts of the liquid.
3. Calculate the total amount of the liquid, to the nearest litre, in the 20 barrels at the end of 20 years.

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.

10. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 4 \\ u _ { n + 1 } & = \frac { 2 u _ { n } } { 3 } , \quad n \geqslant 1 \end{aligned}$$

1. Find the exact values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. Find the value of \(u _ { 20 }\), giving your answer to 3 significant figures.
3. Evaluate $$12 - \sum _ { i = 1 } ^ { 16 } u _ { i }$$ giving your answer to 3 significant figures.
4. Explain why \(\sum _ { i = 1 } ^ { N } u _ { i } < 12\) for all positive integer values of \(N\).

1. The first term of a geometric series is 6 and the common ratio is 0.92

For this series, find

1. 1. the \(25 ^ { \text {th } }\) term, giving your answer to 2 significant figures,
   2. the sum to infinity. The sum to \(n\) terms of this series is greater than 72
2. Calculate the smallest possible value of \(n\).
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11. Wheat is to be grown on a farm. A model predicts that the mass of wheat harvested on the farm will increase by \(1.5 \%\) per year, so that the mass of wheat harvested each year forms a geometric sequence. Given that the mass of wheat harvested during year one is 6000 tonnes,

1. show that, according to the model, the mass of wheat harvested on the farm during year 4 will be approximately 6274 tonnes. During year \(N\), according to the model, there is predicted to be more than 8000 tonnes of wheat harvested on the farm.
2. Find the smallest possible value of \(N\). It costs \(\pounds 5\) per tonne to harvest the wheat.
3. Assuming the model, find the total amount that it would cost to harvest the wheat from year one to year 10 inclusive. Give your answer to the nearest \(\pounds 1000\).

9. A cyclist aims to travel a total of 1200 km over a number of days. He cycles 12 km on day 1
He increases the distance that he cycles each day by \(6 \%\) of the distance cycled on the previous day, until he reaches the total of 1200 km .

1. Show that on day 8 he cycles approximately 18 km . He reaches his total of 1200 km on day \(N\), where \(N\) is a positive integer.
2. Find the value of \(N\). The cyclist stops when he reaches 1200 km .
3. Find the distance that he cycles on day \(N\). Give your answer to the nearest km .

6. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 36 \\ u _ { n + 1 } & = \frac { 2 } { 3 } u _ { n } , \quad n \geqslant 1 \end{aligned}$$

1. Find the exact simplified values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. Write down the common ratio of the sequence.
3. Find, giving your answer to 4 significant figures, the value of \(u _ { 11 }\)
4. Find the exact value of \(\sum _ { i = 1 } ^ { 6 } u _ { i }\)
5. Find the value of \(\sum _ { i = 1 } ^ { \infty } u _ { i }\)

1. A new mineral has been discovered and is going to be mined over a number of years.

A model predicts that the mass of the mineral mined each year will decrease by \(15 \%\) per year, so that the mass of the mineral mined each year forms a geometric sequence. Given that the mass of the mineral mined during year 1 is 8000 tonnes,

1. show that, according to the model, the mass of the mineral mined during year 6 will be approximately 3550 tonnes. According to the model, there is a limit to the total mass of the mineral that can be mined.
2. With reference to the geometric series, state why this limit exists.
3. Calculate the value of this limit. It is decided that a total mass of 40000 tonnes of the mineral is required. This is going to be mined from year 1 to year \(N\) inclusive.
4. Assuming the model, find the value of \(N\).
   

16. The first three terms of a geometric series are \(( k + 5 ) , k\) and \(( 2 k - 24 )\) respectively, where \(k\) is a constant.

1. Show that \(k ^ { 2 } - 14 k - 120 = 0\)
2. Hence find the possible values of \(k\).
3. Given that the series is convergent, find
   1. the common ratio,
   2. the sum to infinity.

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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 ?

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 \(A\).

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.

7. Sue is training for a marathon. Her training includes a run every Saturday starting with a run of 5 km on the first Saturday. Each Saturday she increases the length of her run from the previous Saturday by 2 km .

1. Show that on the 4th Saturday of training she runs 11 km .
2. Find an expression, in terms of \(n\), for the length of her training run on the \(n\)th Saturday.
3. Show that the total distance she runs on Saturdays in \(n\) weeks of training is \(n ( n + 4 ) \mathrm { km }\). On the \(n\)th Saturday Sue runs 43 km .
4. Find the value of \(n\).
5. Find the total distance, in km , Sue runs on Saturdays in \(n\) weeks of training.

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 \(d\).

1. (a) Calculate the sum of all the even numbers from 2 to 100 inclusive,

$$2 + 4 + 6 + \ldots \ldots + 100$$ (b) In the arithmetic series $$k + 2 k + 3 k + \ldots \ldots + 100$$ \(k\) is a positive integer and \(k\) is a factor of 100 .

1. Find, in terms of \(k\), an expression for the number of terms in this series.
2. Show that the sum of this series is $$50 + \frac { 5000 } { k }$$ (c) Find, in terms of \(k\), the 50th term of the arithmetic sequence $$( 2 k + 1 ) , ( 4 k + 4 ) , ( 6 k + 7 ) , \ldots \ldots ,$$ giving your answer in its simplest form.

6. A boy saves some money over a period of 60 weeks. He saves 10 p in week 1 , 15 p in week \(2,20 \mathrm { p }\) in week 3 and so on until week 60 . His weekly savings form an arithmetic sequence.

1. Find how much he saves in week 15
2. Calculate the total amount he saves over the 60 week period. The boy's sister also saves some money each week over a period of \(m\) weeks. She saves 10 p in week \(1,20 \mathrm { p }\) in week \(2,30 \mathrm { p }\) in week 3 and so on so that her weekly savings form an arithmetic sequence. She saves a total of \(\pounds 63\) in the \(m\) weeks.
3. Show that $$m ( m + 1 ) = 35 \times 36$$
4. Hence write down the value of \(m\).

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.