1.04i Geometric sequences: nth term and finite series sum

7. 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 \(N\).

1. Find the value of \(N\). The company then plans to continue to make 600 mobile phones each week.
2. Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.

1. Xin has been given a 14 day training schedule by her coach.

Xin will run for \(A\) minutes on day 1 , where \(A\) is a constant.
She will then increase her running time by ( \(d + 1\) ) minutes each day, where \(d\) is a constant.

1. Show that on day 14 , Xin will run for $$( A + 13 d + 13 ) \text { minutes. }$$ Yi has also been given a 14 day training schedule by her coach.
   Yi will run for \(( A - 13 )\) minutes on day 1 .
   She will then increase her running time by ( \(2 d - 1\) ) minutes each day.
   Given that Yi and Xin will run for the same length of time on day 14,
2. find the value of \(d\). Given that Xin runs for a total time of 784 minutes over the 14 days,
3. find the value of \(A\).

Jess started work 20 years ago. In year 1 her annual salary was \(\pounds 17000\). Her annual salary increased by \(\pounds 1500\) each year, so that her annual salary in year 2 was \(\pounds 18500\), in year 3 it was \(\pounds 20000\) and so on, forming an arithmetic sequence. This continued until she reached her maximum annual salary of \(\pounds 32000\) in year \(k\). Her annual salary then remained at \(\pounds 32000\).

1. Find the value of the constant \(k\).
2. Calculate the total amount that Jess has earned in the 20 years.

9. On John's 10th birthday he received the first of an annual birthday gift of money from his uncle. This first gift was \(\pounds 60\) and on each subsequent birthday the gift was \(\pounds 15\) more than the year before. The amounts of these gifts form an arithmetic sequence.

1. Show that, immediately after his 12th birthday, the total of these gifts was \(\pounds 225\)
2. Find the amount that John received from his uncle as a birthday gift on his 18th birthday.
3. Find the total of these birthday gifts that John had received from his uncle up to and including his 21st birthday. When John had received \(n\) of these birthday gifts, the total money that he had received from these gifts was \(\pounds 3375\)
4. Show that \(n ^ { 2 } + 7 n = 25 \times 18\)
5. Find the value of \(n\), when he had received \(\pounds 3375\) in total, and so determine John's age at this time.

4. A company, which is making 140 bicycles each week, plans to increase its production. The number of bicycles produced is to be increased by \(d\) each week, starting from 140 in week 1 , to \(140 + d\) in week 2 , to \(140 + 2 d\) in week 3 and so on, until the company is producing 206 in week 12.

1. Find the value of \(d\). After week 12 the company plans to continue making 206 bicycles each week.
2. Find the total number of bicycles that would be made in the first 52 weeks starting from and including week 1.

5. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A colony of bees is being studied. The number of bees in the colony at the start of the study was 30000 Three years after the start of the study, the number of bees in the colony is 34000 A model predicts that the number of bees in the colony will increase by \(p \%\) each year, so that the number of bees in the colony at the end of each year of study forms a geometric sequence. Assuming the model,

1. find the value of \(p\), giving your answer to 2 decimal places. According to the model, at the end of \(N\) years of study the number of bees in the colony exceeds 75000
2. Find, showing all steps in your working, the smallest integer value of \(N\).

10. In this question you must show detailed reasoning. Owen wants to train for 12 weeks in preparation for running a marathon. During the 12-week period he will run every Sunday and every Wednesday.

• On Sunday in week 1 he will run 15 km
• On Sunday in week 12 he will run 37 km

He considers two different 12-week training plans. In training plan \(A\), he will increase the distance he runs each Sunday by the same amount.

1. Calculate the distance he will run on Sunday in week 5 under training plan \(A\). In training plan \(B\), he will increase the distance he runs each Sunday by the same percentage.
2. Calculate the distance he will run on Sunday in week 5 under training plan \(B\). Give your answer in km to one decimal place. Owen will also run a fixed distance, \(x \mathrm {~km}\), each Wednesday over the 12-week period. Given that
   • \(x\) is an integer
   • the total distance that Owen will run on Sundays and Wednesdays over the 12 weeks will not exceed 360 km
   • 1. find the maximum value of \(x\), if he uses training plan \(A\),
     2. find the maximum value of \(x\), if he uses training plan \(B\).
        
   \includegraphics[max width=\textwidth, alt={}, center]{52c90d0e-a5e4-45fa-95a4-9523287e7588-31_2255_50_314_34}

8. A metal post is repeatedly hit in order to drive it into the ground. Given that

• on the 1st hit, the post is driven 100 mm into the ground
• on the 2nd hit, the post is driven an additional 98 mm into the ground
• on the 3rd hit, the post is driven an additional 96 mm into the ground
• the additional distances the post travels on each subsequent hit form an arithmetic sequence
  1. show that the post is driven an additional 62 mm into the ground with the 20th hit.
  2. Find the total distance that the post has been driven into the ground after 20 hits.

Given that for each subsequent hit after the 20th hit

After a total of \(N\) hits, the post will have been driven more than 3 m into the ground.

Find, showing all steps in your working, the smallest possible value of \(N\).

1. A geometric sequence has first term \(a\) and common ratio \(r\), where \(r > 0\)

Given that

• the 3rd term is 20
• the 5th term is 12.8
  1. show that \(r = 0.8\)
  2. Hence find the value of \(a\).

Given that the sum of the first \(n\) terms of this sequence is greater than 156

find the smallest possible value of \(n\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)

1. Wheat is grown on a farm.

• In year 1 , the farm produced 300 tonnes of wheat.
• In year 12 , the farm is predicted to produce 4000 tonnes of wheat.

Model \(A\) assumes that the amount of wheat produced on the farm will increase by the same amount each year.

1. Using model \(A\), find the amount of wheat produced on the farm in year 4. Give your answer to the nearest 10 tonnes. Model \(B\) assumes that the amount of wheat produced on the farm will increase by the same percentage each year.
2. Using model \(B\), find the amount of wheat produced on the farm in year 2. Give your answer to the nearest 10 tonnes.
3. Calculate, according to the two models, the difference between the total amounts of wheat predicted to be produced on the farm from year 1 to year 12 inclusive. Give your answer to the nearest 10 tonnes.

7. Kim starts working for a company.

• In year 1 her annual salary will be \(\pounds 16200\)
• In year 10 her annual salary is predicted to be \(\pounds 31500\)

Model \(A\) assumes that her annual salary will increase by the same amount each year.

1. According to model \(A\), determine Kim's annual salary in year 2 . Model \(B\) assumes that her annual salary will increase by the same percentage each year.
2. According to model \(B\), determine Kim's annual salary in year 2 . Give your answer to the nearest \(\pounds 10\)
3. Calculate, according to the two models, the difference between the total amounts that Kim is predicted to earn from year 1 to year 10 inclusive. Give your answer to the nearest £10

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A software developer released an app to download.
The numbers of downloads of the app each month, in thousands, for the first three months after the app was released were $$2 k - 15 \quad k \quad k + 4$$ where \(k\) is a constant.
Given that the numbers of downloads each month are modelled as a geometric series,

1. show that \(k ^ { 2 } - 7 k - 60 = 0\)
2. predict the number of downloads in the 4th month. The total number of all downloads of the app is predicted to exceed 3 million for the first time in the \(N\) th month.
3. Calculate the value of \(N\) according to the model.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The number of dormice and the number of voles on an island are being monitored.
Initially there are 2000 dormice on the island.
A model predicts that the number of dormice will increase by \(3 \%\) each year, so that the numbers of dormice on the island at the end of each year form a geometric sequence.

1. Find, according to the model, the number of dormice on the island 6 years after monitoring began. Give your answer to 3 significant figures. The number of voles on the island is being monitored over the same period of time.
   Given that
   • 4 years after monitoring began there were 3690 voles on the island
   • 7 years after monitoring began there were 3470 voles on the island
   • the number of voles on the island at the end of each year is modelled as a geometric sequence
   • find the equation of this model in the form
   $$N = a b ^ { t }$$ where \(N\) is the number of voles, \(t\) years after monitoring began and \(a\) and \(b\) are constants. Give the value of \(a\) and the value of \(b\) to 2 significant figures. When \(t = T\), the number of dormice on the island is equal to the number of voles on the island.
2. Find, according to the models, the value of \(T\), giving your answer to one decimal place.

2. The adult population of a town at the start of 2019 is 25000 A model predicts that the adult population will increase by \(2 \%\) each year, so that the number of adults in the population at the start of each year following 2019 will form a geometric sequence.

1. Find, according to the model, the adult population of the town at the start of 2032 It is also modelled that every member of the adult population gives \(\pounds 5\) to local charity at the start of each year.
2. Find, according to these models, the total amount of money that would be given to local charity by the adult population of the town from 2019 to 2032 inclusive. Give your answer to the nearest \(\pounds 1000\)

5. Ben is saving for the deposit for a house over a period of 60 months. Ben saves \(\pounds 100\) in the first month and in each subsequent month, he saves \(\pounds 5\) more than the previous month, so that he saves \(\pounds 105\) in the second month, \(\pounds 110\) in the third month, and so on, forming an arithmetic sequence.

1. Find the amount Ben saves in the 40th month.
2. Find the total amount Ben saves over the 60 -month period. Lina is also saving for a deposit for a house.
   Lina saves \(\pounds 600\) in the first month and in each subsequent month, she saves \(\pounds 10\) less than the previous month, so that she saves \(\pounds 590\) in the second month, \(\pounds 580\) in the third month, and so on, forming an arithmetic sequence. Given that, after \(n\) months, Lina will have saved exactly \(\pounds 18200\) for her deposit,
3. form an equation in \(n\) and show that it can be written as $$n ^ { 2 } - 121 n + 3640 = 0$$
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 value for \(n\).

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

1. Prove that the sum of the first \(n\) terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ The second term of a geometric series is - 320 and the fifth term is \(\frac { 512 } { 25 }\)
2. Find the value of the common ratio.
3. Hence find the sum of the first 13 terms of the series, giving your answer to 2 decimal places.

5. A company that owned a silver mine

• extracted 480 tonnes of silver from the mine in year 1
• extracted 465 tonnes of silver from the mine in year 2
• extracted 450 tonnes of silver from the mine in year 3
  and so on, forming an arithmetic sequence.
  1. Find the mass of silver extracted in year 14

After a total of 7770 tonnes of silver was extracted, the company stopped mining. Given that this occurred at the end of year \(N\),

show that $$N ^ { 2 } - 65 N + 1036 = 0$$

Hence, state the value of \(N\).

7. (i) A geometric sequence has first term 4 and common ratio 6 Given that the \(n ^ { \text {th } }\) term is greater than \(10 ^ { 100 }\), find the minimum possible value of \(n\).
(ii) A different geometric sequence has first term \(a\) and common ratio \(r\). Given that

• the second term of the sequence is - 6
• the sum to infinity of the series is 25
  1. show that

$$25 r ^ { 2 } - 25 r - 6 = 0$$

Write down the solutions of $$25 r ^ { 2 } - 25 r - 6 = 0$$ Hence,

state the value of \(r\), giving a reason for your answer,

find the sum of the first 4 terms of the series. \includegraphics[max width=\textwidth, alt={}, center]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-23_70_37_2617_1914}

1. A geometric sequence has first term \(a\) and common ratio \(r\)

Given that \(S _ { \infty } = 3 a\)

1. show that \(r = \frac { 2 } { 3 }\) Given also that $$u _ { 2 } - u _ { 4 } = 16$$ where \(u _ { k }\) is the \(k ^ { \text {th } }\) term of this sequence,
2. find the value of \(S _ { 10 }\) giving your answer to one decimal place.

2. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\). The sum to infinity of the series is \(S _ { \infty }\)

1. Find the value of \(S _ { \infty }\) The sum to \(N\) terms of the series is \(S _ { N }\)
2. Find, to 1 decimal place, the value of \(S _ { 12 }\)
3. Find the smallest value of \(N\), for which \(S _ { \infty } - S _ { N } < 0.5\) 2. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\). The sum to infinity
   of the series is \(S _ { \infty }\)

1. The second and fourth terms of a geometric series are 7.2 and 5.832 respectively.

The common ratio of the series is positive.
For this series, find

1. the common ratio,
2. the first term,
3. the sum of the first 50 terms, giving your answer to 3 decimal places,
4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places.

6. A car was purchased for \(\pounds 18000\) on 1 st January. On 1st January each following year, the value of the car is \(80 \%\) of its value on 1st January in the previous year.

1. Show that the value of the car exactly 3 years after it was purchased is \(\pounds 9216\). The value of the car falls below \(\pounds 1000\) for the first time \(n\) years after it was purchased.
2. Find the value of \(n\). An insurance company has a scheme to cover the maintenance of the car. The cost is \(\pounds 200\) for the first year, and for every following year the cost increases by \(12 \%\) so that for the 3rd year the cost of the scheme is \(\pounds 250.88\)
3. Find the cost of the scheme for the 5th year, giving your answer to the nearest penny.
4. Find the total cost of the insurance scheme for the first 15 years.
   \section*{LU}

3. The second and fifth terms of a geometric series are 750 and - 6 respectively. Find

1. the common ratio of the series,
2. the first term of the series,
3. the sum to infinity of the series.