1.04j Sum to infinity: convergent geometric series |r|<1

5 The sum of the 1st and 2nd terms of a geometric progression is 50 and the sum of the 2nd and 3rd terms is 30 . Find the sum to infinity.

8

1. A cyclist completes a long-distance charity event across Africa. The total distance is 3050 km . He starts the event on May 1st and cycles 200 km on that day. On each subsequent day he reduces the distance cycled by 5 km .
   1. How far will he travel on May 15th?
   2. On what date will he finish the event?
2. A geometric progression is such that the third term is 8 times the sixth term, and the sum of the first six terms is \(31 \frac { 1 } { 2 }\). Find
   1. the first term of the progression,
   2. the sum to infinity of the progression.

3

1. A geometric progression has first term \(3 a\) and common ratio \(r\). A second geometric progression has first term \(a\) and common ratio \(- 2 r\). The two progressions have the same sum to infinity. Find the value of \(r\).
2. The first two terms of an arithmetic progression are 15 and 19 respectively. The first two terms of a second arithmetic progression are 420 and 415 respectively. The two progressions have the same sum of the first \(n\) terms. Find the value of \(n\).

9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 8 \\ - 6 \\ 5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 10 \\ 3 \\ - 13 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2 \\ - 3 \\ - 1 \end{array} \right)$$ A fourth point, \(D\), is such that the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\) are the first, second and third terms respectively of a geometric progression.

1. Find the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\).
2. Given that \(D\) is a point lying on the line through \(B\) and \(C\), find the two possible position vectors of the point \(D\).

4 The first term of a series is 6 and the second term is 2 .

1. For the case where the series is an arithmetic progression, find the sum of the first 80 terms.
2. For the case where the series is a geometric progression, find the sum to infinity.

8

1. Over a 21-day period an athlete prepares for a marathon by increasing the distance she runs each day by 1.2 km . On the first day she runs 13 km .
   1. Find the distance she runs on the last day of the 21-day period.
   2. Find the total distance she runs in the 21-day period.
2. The first, second and third terms of a geometric progression are \(x , x - 3\) and \(x - 5\) respectively.
   1. Find the value of \(x\).
   2. Find the fourth term of the progression.
   3. Find the sum to infinity of the progression.

9 The first, second and third terms of a geometric progression are \(3 k , 5 k - 6\) and \(6 k - 4\), respectively.

1. Show that \(k\) satisfies the equation \(7 k ^ { 2 } - 48 k + 36 = 0\).
2. Find, showing all necessary working, the exact values of the common ratio corresponding to each of the possible values of \(k\).
3. One of these ratios gives a progression which is convergent. Find the sum to infinity.

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.

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 \(N\).

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 \(a\).

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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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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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 \(N\).

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. 1. Find the value of

$$\sum _ { r = 1 } ^ { \infty } 6 \times ( 0.25 ) ^ { r }$$ (3)
(ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$\begin{aligned} u _ { 1 } & = 3 \\ u _ { n + 1 } & = \frac { u _ { n } - 3 } { u _ { n } - 2 } \quad n \in \mathbb { N } \end{aligned}$$

1. Show that this sequence is periodic.
2. State the order of this sequence.
3. Hence find $$\sum _ { n = 1 } ^ { 70 } u _ { n }$$

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.

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 \(\pounds 1000\)

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.

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.