1.04i Geometric sequences: nth term and finite series sum

7 Chris runs half marathons, and is following a training programme to improve his times. His time for his first half marathon is 150 minutes. His time for his second half marathon is 147 minutes. Chris believes that his times can be modelled by a geometric progression.

1. Chris sets himself a target of completing a half marathon in less than 120 minutes. Show that this model predicts that Chris will achieve his target on his thirteenth half marathon.
2. After twelve months Chris has spent a total of 2974 minutes, to the nearest minute, running half marathons. Use this model to find how many half marathons he has run.
3. Give two reasons why this model may not be appropriate when predicting the time for a half marathon.

7 Two students, Anna and Ben, are starting a revision programme. They will both revise for 30 minutes on Day 1. Anna will increase her revision time by 15 minutes for every subsequent day. Ben will increase his revision time by \(10 \%\) for every subsequent day.

1. Verify that on Day 10 Anna does 94 minutes more revision than Ben, correct to the nearest minute. Let Day \(X\) be the first day on which Ben does more revision than Anna.
2. Show that \(X\) satisfies the inequality \(X > \log _ { 1.1 } ( 0.5 X + 0.5 ) + 1\).
3. Use the iterative formula \(x _ { n + 1 } = \log _ { 1.1 } \left( 0.5 x _ { n } + 0.5 \right) + 1\) with \(x _ { 1 } = 10\) to find the value of \(X\). You should show the result of each iteration.
4. 1. Give a reason why Anna's revision programme may not be realistic.
   2. Give a different reason why Ben's revision programme may not be realistic.

1. Ben starts a new company.

• In year 1 his profits will be \(\pounds 24000\).
• In year 11 his profit is predicted to be \(\pounds 64000\).

Model \(\boldsymbol { P }\) assumes that his profit will increase by the same amount each year.
a. According to model \(\boldsymbol { P }\), determine Ben's profit in year 5. Model \(\boldsymbol { Q }\) assumes that his profit will increase by the same percentage each year.
b. According to model \(\boldsymbol { Q }\), determine Ben's profit in year 5 . Give your answer to the nearest £10.

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 \(\sum _ { r = 1 } ^ { 222 } u _ { r }\)

15. The first three terms of a geometric series where \(\theta\) is a constant are $$- 8 \sin \theta , \quad 3 - 2 \cos \theta \quad \text { and } \quad 4 \cot \theta$$ a. Show that \(4 \cos ^ { 2 } \theta + 20 \cos \theta + 9 = 0\) Given that \(\theta\) lies in the interval \(90 ^ { \circ } < \theta < 180 ^ { \circ }\),
b. Find the value of \(\theta\).
c. Hence prove that this series is convergent.
d. Find \(S _ { \infty }\), in the form \(a ( 1 - \sqrt { 3 } )\)

1. A competitor is running a 20 kilometre race.

She runs each of the first 4 kilometres at a steady pace of 6 minutes per kilometre. After the first 4 kilometres, she begins to slow down. In order to estimate her finishing time, the time that she will take to complete each subsequent kilometre is modelled to be \(5 \%\) greater than the time that she took to complete the previous kilometre. Using the model,

1. show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds,
2. show that her estimated time, in minutes, to run the \(r\) th kilometre, for \(5 \leqslant r \leqslant 20\), is $$6 \times 1.05 ^ { r - 4 }$$
3. estimate the total time, in minutes and seconds, that she will take to complete the race.

1. The first three terms of a geometric sequence are

$$3 k + 4 \quad 12 - 3 k \quad k + 16$$ where \(k\) is a constant.

1. Show that \(k\) satisfies the equation $$3 k ^ { 2 } - 62 k + 40 = 0$$ Given that the sequence converges,
2. 1. find the value of \(k\), giving a reason for your answer,
   2. find the value of \(S _ { \infty }\)

1. The first 3 terms of a geometric sequence are

$$3 ^ { 4 k - 5 } \quad 9 ^ { 7 - 2 k } \quad 3 ^ { 2 ( k - 1 ) }$$ where \(k\) is a constant.

1. Using algebra and making your reasoning clear, prove that \(k = \frac { 5 } { 2 }\)
2. Hence find the sum to infinity of the geometric sequence.

1. A car has six forward gears.

The fastest speed of the car

• in \(1 ^ { \text {st } }\) gear is \(28 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
• in \(6 ^ { \text {th } }\) gear is \(115 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)

Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence,

1. find the fastest speed of the car in \(3 { } ^ { \text {rd } }\) gear. Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,
2. find the fastest speed of the car in \(5 ^ { \text {th } }\) gear.

1. The curve with equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = x ^ { 2 } + \ln \left( 2 x ^ { 2 } - 4 x + 5 \right)$$ has a single turning point at \(x = \alpha\)

1. Show that \(\alpha\) is a solution of the equation $$2 x ^ { 3 } - 4 x ^ { 2 } + 7 x - 2 = 0$$ The iterative formula $$x _ { n + 1 } = \frac { 1 } { 7 } \left( 2 + 4 x _ { n } ^ { 2 } - 2 x _ { n } ^ { 3 } \right)$$ is used to find an approximate value for \(\alpha\).
   Starting with \(x _ { 1 } = 0.3\)
2. calculate, giving each answer to 4 decimal places,
   1. the value of \(x _ { 2 }\)
   2. the value of \(x _ { 4 }\) Using a suitable interval and a suitable function that should be stated,
3. show that \(\alpha\) is 0.341 to 3 decimal places.

1. There were 2100 tonnes of wheat harvested on a farm during 2017.

The mass of wheat harvested during each subsequent year is expected to increase by \(1.2 \%\) per year.

1. Find the total mass of wheat expected to be harvested from 2017 to 2030 inclusive, giving your answer to 3 significant figures. Each year it costs
   • £5.15 per tonne to harvest the first 2000 tonnes of wheat
   • £6.45 per tonne to harvest wheat in excess of 2000 tonnes
   • Use this information to find the expected cost of harvesting the wheat from 2017 to 2030 inclusive. Give your answer to the nearest \(\pounds 1000\)

1. (i) Show that \(\sum _ { r = 1 } ^ { 16 } \left( 3 + 5 r + 2 ^ { r } \right) = 131798\) (ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by

$$u _ { n + 1 } = \frac { 1 } { u _ { n } } , \quad u _ { 1 } = \frac { 2 } { 3 }$$ Find the exact value of \(\sum _ { r = 1 } ^ { 100 } u _ { r }\)

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.} Given that the first three terms of a geometric series are $$12 \cos \theta \quad 5 + 2 \sin \theta \quad \text { and } \quad 6 \tan \theta$$

1. show that $$4 \sin ^ { 2 } \theta - 52 \sin \theta + 25 = 0$$ Given that \(\theta\) is an obtuse angle measured in radians,
2. solve the equation in part (a) to find the exact value of \(\theta\)
3. show that the sum to infinity of the series can be expressed in the form $$k ( 1 - \sqrt { 3 } )$$ where \(k\) is a constant to be found.

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} A geometric series has common ratio \(r\) and first term \(a\).
Given \(r \neq 1\) and \(a \neq 0\)

1. prove that $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ Given also that \(S _ { 10 }\) is four times \(S _ { 5 }\)
2. find the exact value of \(r\).

10. In a geometric series the common ratio is \(r\) and sum to \(n\) terms is \(S _ { n }\) Given $$S _ { \infty } = \frac { 8 } { 7 } \times S _ { 6 }$$ show that \(r = \pm \frac { 1 } { \sqrt { k } }\), where \(k\) is an integer to be found.

6 Aleela and Baraka are saving to buy a car. Aleela saves \(\pounds 50\) in the first month. She increases the amount she saves by \(\pounds 20\) each month.

1. Calculate how much she saves in two years. Baraka also saves \(\pounds 50\) in the first month. The amount he saves each month is \(12 \%\) more than the amount he saved in the previous month.
2. Explain why the amounts Baraka saves each month form a geometric sequence.
3. Determine whether Baraka saves more in two years than Aleela. Answer all the questions
   Section B (77 marks)

11 The first three terms of a geometric sequence are \(5 k - 2,3 k - 6 , k + 2\), where \(k\) is a constant.

1. Show that \(k\) satisfies the equation \(k ^ { 2 } - 11 k + 10 = 0\).
2. When \(k\) takes the smaller of the two possible values, find the sum of the first 20 terms of the sequence.
3. When \(k\) takes the larger of the two possible values, find the sum to infinity of the sequence.

4

1. The first four terms of a sequence are \(2,3,0,3\) and the subsequent terms are given by \(\mathrm { a } _ { \mathrm { k } + 4 } = \mathrm { a } _ { \mathrm { k } }\).
   1. State what type of sequence this is.
   2. Find \(\sum _ { \mathrm { k } = 1 } ^ { 200 } \mathrm { a } _ { \mathrm { k } }\).
2. A different sequence is given by \(\mathrm { u } _ { \mathrm { n } } = \mathrm { b } ^ { \mathrm { n } }\) where \(b\) is a constant and \(n \geqslant 1\).
   1. State the set of values of \(b\) for which this is a divergent sequence.
   2. In the case where \(b = \frac { 1 } { 3 }\), find the sum of all the terms in the sequence.

13 The population of Melchester is 185207. During a nationwide flu epidemic the number of new cases in Melchester are recorded each day. The results from the first three days are shown in Fig. 13. \begin{table}[h] \end{table} A doctor notices that the numbers of new cases on successive days are in geometric progression.

1. Find the common ratio for this geometric progression. The doctor uses this geometric progression to model the number of new cases of flu in Melchester.
2. According to the model, how many new cases will there be on day 5?
3. Find a formula for the total number of cases from day 1 to day \(n\) inclusive according to this model, simplifying your answer.
4. Determine the maximum number of days for which the model could be viable in Melchester.
5. State, with a reason, whether it is likely that the model will be viable for the number of days found in part (d).

13 Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three consecutive terms of this sequence, the middle one is the geometric mean of the other two.

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.

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 .

3

1. Use logarithms to solve the equation \(0.8 ^ { x } = 0.05\), giving your answer to three decimal places.
2. An infinite geometric series has common ratio \(r\). The sum to infinity of the series is five times the first term of the series.
   1. Show that \(r = 0.8\).
   2. Given that the first term of the series is 20 , find the least value of \(n\) such that the \(n\)th term of the series is less than 1 .

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 .