1.04i Geometric sequences: nth term and finite series sum

9. The second and fifth terms of a geometric series are - 48 and 6 respectively.

1. Find the first term and the common ratio of the series.
2. Find the sum to infinity of the series.
3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2 ^ { 6 - n }\).

7 Business A made a \(\pounds 5000\) profit during its first year.
In each subsequent year, the profit increased by \(\pounds 1500\) so that the profit was \(\pounds 6500\) during the second year, \(\pounds 8000\) during the third year and so on. Business B made a \(\pounds 5000\) profit during its first year.
In each subsequent year, the profit was 90\% of the previous year's profit.

1. Find an expression for the total profit made by business A during the first \(n\) years. Give your answer in its simplest form.
2. Find an expression for the total profit made by business B during the first \(n\) years. Give your answer in its simplest form.
3. Find how many years it will take for the total profit of business A to reach \(\pounds 385000\).
4. Comment on the profits made by each business in the long term.

4 An artist is creating a design for a large painting. The design includes a set of steps of varying heights. In the painting the lowest step has height 20 cm and the height of each other step is \(5 \%\) less than the height of the step immediately below it. In the painting the total height of the steps is 205 cm , correct to the nearest centimetre. Determine the number of steps in the design.

14

1. 1. Given that $$y = 2 ^ { x }$$ write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) 14
      1. (ii) Hence find $$\int 2 ^ { x } \mathrm {~d} x$$ 14
   2. The area, \(A\), bounded by the curve with equation \(y = 2 ^ { x }\), the \(x\)-axis, the \(y\)-axis and the line \(x = - 4\) is approximated using eight rectangles of equal width as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-23_1319_978_450_532} 14
   3. 1. Show that the exact area of the largest rectangle is \(\frac { \sqrt { 2 } } { 4 }\) 14
   4. (ii) The areas of these rectangles form a geometric sequence with common ratio \(\frac { \sqrt { 2 } } { 2 }\) Find the exact value of the total area of the eight rectangles.
      Give your answer in the form \(k ( 1 + \sqrt { 2 } )\) where \(k\) is a rational number.
      [0pt] [3 marks]
      14
   5. (iii) More accurate approximations for \(A\) can be found by increasing the number, \(n\), of rectangles used. Find the exact value of the limit of the approximations for \(A\) as \(n \rightarrow \infty\)

5 Ziad is training to become a long-distance swimmer. He trains every day by swimming lengths at his local pool.
The length of the pool is 25 metres.
Each day he increases the number of lengths that he swims by four.
On his first day of training, Ziad swims 10 lengths of the pool.
5

1. Write down an expression for the number of lengths Ziad will swim on his \(n\)th day of training. 5
2. (i) Ziad's target is to be able to swim at least 3000 metres in one day.
   Determine the minimum number of days he will need to train to reach his target.
   5 (b) (ii) Ziad's coach claims that when he reaches his target he will have covered a total distance of over 50000 metres. Determine if Ziad's coach is correct.

7 In a long-running biochemical experiment, an initial amount of 1200 mg of an enzyme is placed into a mixture. The model for the amount of enzyme present in the mixture suggests that, at the end of each hour, one-eighth of the amount of enzyme that was present at the start of that hour is used up due to chemical reactions within the mixture. To compensate for this, at the end of each six-hour period of time, a further 500 mg of the enzyme is added to the mixture.

1. Let \(n\) be the number of six-hour periods that have elapsed since the experiment began. Explain how the amount of enzyme, \(\mathrm { E } _ { \mathrm { n } } \mathrm { mg }\), in the mixture is given by the recurrence system \(E _ { 0 } = 1200\) and \(E _ { n + 1 } = \left( \frac { 7 } { 8 } \right) ^ { 6 } E _ { n } + 500\) for \(n \geqslant 0\).
2. Solve the recurrence system given in part (a) to obtain an exact expression for \(\mathrm { E } _ { \mathrm { n } }\) in terms of \(n\).
3. Hence determine, in the long term, the amount of enzyme in the mixture. Give your answer correct to \(\mathbf { 3 }\) significant figures.
4. In this question you must show detailed reasoning. The long-running experiment is then repeated. This time a new requirement is added that the amount of enzyme present in the mixture must always be at least 500 mg . Show that the new requirement ceases to be satisfied before 12 hours have elapsed. \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
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1. Jem pays money into a savings scheme, \(A\), over a period of 300 months.

Jem pays \(\pounds 20\) into scheme \(A\) in month \(1 , \pounds 20.50\) in month \(2 , \pounds 21\) in month 3 and so on, so that the amounts Jem pays each month form an arithmetic sequence.

1. Show that Jem pays \(\pounds 69.50\) into scheme \(A\) in month 100
2. Find the total amount that Jem pays into scheme \(A\) over the period of 300 months. Kim pays money into a different savings scheme, \(B\), over the same period of 300 months. In a model, the amounts Kim pays into scheme \(B\) increase by the same percentage each month, so that the amounts Kim pays each month form a geometric sequence. Given that Kim pays
   • \(\pounds 20\) into scheme \(B\) in month 1
   • \(\pounds 250\) into scheme \(B\) in month 300
   • use the model to calculate, to the nearest \(\pounds 10\), the difference between the total amount paid into scheme \(A\) and the total amount paid into scheme \(B\) over the period of 300 months.

1 The first term of a geometric progression is 16 and the common ratio is 0.8 .

1. Calculate the sum of the first 12 terms.
2. Find the sum to infinity.

3 The first term of a geometric progression is 50 and the common ratio is 0.9 .

1. Find the fifth term.
2. Find the sum of the first thirty terms.
3. Find the sum to infinity.

9 It is given that \(x , 6\) and \(x + 5\) are consecutive terms of a geometric progression.

1. Show that \(x ^ { 2 } + 5 x - 36 = 0\) and find the possible values of \(x\).
2. Hence find the possible values of the common ratio. Furthermore, \(x , 6\) and \(x + 5\) are the second, third and fourth terms of a geometric progression for which the sum to infinity exists.
3. Find the first term and the sum to infinity.

11 An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).

1. Find \(d\) in terms of \(a\) and show that \(r = \frac { 5 } { 8 }\).
2. Find the sum to infinity of the geometric progression in terms of \(a\).

10 An arithmetic sequence and a geometric sequence have \(n\)th terms \(a _ { n }\) and \(g _ { n }\) respectively, where \(n = 1,2,3 , \ldots\). It is given that \(a _ { 1 } = g _ { 1 } , a _ { 2 } = g _ { 2 } , a _ { 5 } = g _ { 3 } , a _ { 1 } \neq a _ { 2 }\) and \(a _ { 1 } \neq 0\).

1. Show that the common ratio of the geometric sequence is 3 .
2. Find the common difference of the arithmetic sequence in terms of \(a _ { 1 }\).
3. Let \(a _ { 1 } = g _ { 1 } = 5\).
   1. Find the first three terms of both sequences.
   2. Show that every term of the geometric sequence is also a term of the arithmetic sequence.

1 A geometric progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 32\) and \(u _ { n + 1 } = 0.75 u _ { n }\) for \(n \geqslant 1\).

1. Find \(u _ { 5 }\).
2. Find \(\sum _ { n = 1 } ^ { \infty } u _ { n }\).

11 An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).

1. Find \(d\) in terms of \(a\) and show that \(r = \frac { 5 } { 8 }\).
2. Find the sum to infinity of the geometric progression in terms of \(a\).

6

1. Given that the numbers \(a , b\) and \(c\) are in arithmetic progression, show that \(a + c = 2 b\).
2. Find an analogous result for three numbers in geometric progression.
3. The numbers \(2 - 3 x , 2 x , 3 - 2 x\) are the first three terms of a convergent geometric progression. Find \(x\) and hence calculate the sum to infinity.

8

1. The sum of the first \(n\) terms of the arithmetic series \(1 + 3 + 5 + \ldots\) exceeds the sum of the first \(n\) terms of the arithmetic series \(100 + 97 + 94 + \ldots\). Find the least possible value of \(n\).
2. \(3 \sqrt { 2 }\) and \(2 - \sqrt { 2 }\) are the first two terms of a geometric progression.
   1. Show that the third term is \(\sqrt { 2 } - \frac { 4 } { 3 }\).
   2. Find the index \(n\) of the first term that is less than 0.01.
   3. Find the exact value of the sum to infinity of this progression.
   4. Which of the terms 'alternating', 'periodic', 'convergent' apply to the sequences generated by the following \(n\)th terms, where \(n\) is a positive integer?
      (a) \(1 - \left( \frac { 3 } { 4 } \right) ^ { n }\) (b) \(\frac { 1 } { n } \cos n \pi\) (c) \(\sec n \pi\)

a) The \(3 ^ { \text {rd } } , 19 ^ { \text {th } }\) and \(67 ^ { \text {th } }\) terms of an arithmetic sequence form a geometric sequence. Given that the arithmetic sequence is increasing and that the first term is 3 , find the common difference of the arithmetic sequence. b) A firm has 100 employees on a particular Monday. The next day it adds 12 employees onto its staff and continues to do so on every successive working day, from Monday to Friday.
i) Find the number of employees at the end of the \(8 { } ^ { \text {th } }\) week.
ii) Each employee is paid \(\pounds 55\) per working day. Determine the total wage bill for the 8 week period.

The second term of a geometric progression is 16 and the sum to infinity is 100.

1. Find the two possible values of the first term. [4]
2. Show that the \(n\)th term of one of the two possible geometric progressions is equal to \(4^{n-2}\) multiplied by the \(n\)th term of the other geometric progression. [4]

The first and second terms of an arithmetic progression are \(\tan\theta\) and \(\sin\theta\) respectively, where \(0 < \theta < \frac{1}{2}\pi\). \begin{enumerate}[label=(\alph*)] \item Given that \(\theta = \frac{1}{4}\pi\), find the exact sum of the first 40 terms of the progression. [4] \end enumerate} The first and second terms of a geometric progression are \(\tan\theta\) and \(\sin\theta\) respectively, where \(0 < \theta < \frac{1}{2}\pi\).

1. 1. Find the sum to infinity of the progression in terms of \(\theta\). [2]
   2. Given that \(\theta = \frac{1}{3}\pi\), find the sum of the first 10 terms of the progression. Give your answer correct to 3 significant figures. [3]

The first, second and third terms of a geometric progression are \(\sin\theta\), \(\cos\theta\) and \(2 - \sin\theta\) respectively, where \(\theta\) radians is an acute angle.

1. Find the value of \(\theta\). [3]
2. Using this value of \(\theta\), find the sum of the first 10 terms of the progression. Give the answer in the form \(\frac{b}{\sqrt{c} - 1}\), where \(b\) and \(c\) are integers to be found. [3]

An arithmetic progression has first term 5 and common difference \(d\), where \(d > 0\). The second, fifth and eleventh terms of the arithmetic progression, in that order, are the first three terms of a geometric progression.

1. Find the value of \(d\). [3]
2. The sum of the first 77 terms of the arithmetic progression is denoted by \(S_{77}\). The sum of the first 10 terms of the geometric progression is denoted by \(G_{10}\). Find the value of \(S_{77} - G_{10}\). [5]

The first term of a convergent geometric progression is 10. The sum of the first 4 terms of the progression is \(p\) and the sum of the first 8 terms of the progression is \(q\). It is given that \(\frac{q}{p} = \frac{17}{16}\). Find the two possible values of the sum to infinity. [5]