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

6 In this question you must show detailed reasoning.
A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) defined by \(u _ { 1 } = 500\) and \(u _ { n + 1 } = 0.8 u _ { n }\).

1. State whether \(S\) is an arithmetic sequence or a geometric sequence, giving a reason for your answer.
2. Find \(u _ { 20 }\).
3. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
4. Given that \(\sum _ { n = k } ^ { \infty } u _ { n } = 1024\), find the value of \(k\).

5 The second term of a geometric series is 48 and the fourth term is 3 .

1. Show that one possible value for the common ratio, \(r\), of the series is \(- \frac { 1 } { 4 }\) and state the other value.
2. In the case when \(r = - \frac { 1 } { 4 }\), find:
   1. the first term;
   2. the sum to infinity of the series.

2 The \(n\)th term of a geometric sequence is \(u _ { n }\), where $$u _ { n } = 3 \times 4 ^ { n }$$

1. Find the value of \(u _ { 1 }\) and show that \(u _ { 2 } = 48\).
2. Write down the common ratio of the geometric sequence.
3. 1. Show that the sum of the first 12 terms of the geometric sequence is \(4 ^ { k } - 4\), where \(k\) is an integer.
   2. Hence find the value of \(\sum _ { n = 2 } ^ { 12 } u _ { n }\).

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.

3 A geometric sequence has a sum to infinity of - 3 A second sequence is formed by multiplying each term of the original sequence by - 2 What is the sum to infinity of the new sequence? Circle your answer. The sum to infinity does not

12

1. A geometric sequence has first term 1 and common ratio \(\frac { 1 } { 2 }\) 12
   1. (i) Find the sum to infinity of the sequence.
      12
   2. (ii) Hence, or otherwise, evaluate $$\sum _ { n = 1 } ^ { \infty } \left( \sin 30 ^ { \circ } \right) ^ { n }$$ 12
   3. Find the smallest positive exact value of \(\theta\), in radians, which satisfies the equation $$\sum _ { n = 0 } ^ { \infty } ( \cos \theta ) ^ { n } = 2 - \sqrt { 2 }$$

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 \(n \rightarrow \infty\)

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 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

1. In an arithmetic progression, the first term is 7 and the sum of the first 40 terms is 4960. Find the common difference.
2. A geometric progression has first term 14 and common ratio 0.3. Find the sum to infinity.

2

1. An arithmetic sequence has first term 3 and common difference 2. Find the twenty-first term of this sequence.
2. Find the sum to infinity of a geometric progression with first term 162 and second term 54.
3. A sequence is given by the recurrence relation \(u _ { 1 } = 3 , u _ { n + 1 } = 2 - u _ { n } , n = 1,2,3 , \ldots\). Find \(u _ { 2 } , u _ { 3 }\), \(u _ { 4 } , u _ { 5 }\) and describe the behaviour of this sequence.

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 \(a\).

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) \(\sec n \pi\)

The diagram below shows a plan of the patio Eric wants to build. The walls \(O A\) and \(O C\) are perpendicular. The straight line \(A B\) is of length 4 m and is perpendicular to \(O A\). The shape \(O B C\) is a sector of a circle with centre \(O\) and radius OC.
The angle \(B O C\) is \(\frac { \pi } { 3 }\) radians. Calculate the area of the patio \(O A B C\). Give your answer correct to 2 decimal places. The sum to infinity of a geometric series with first term \(a\) and common ratio \(r\) is 120 . The sum to infinity of another geometric series with first term \(a\) and common ratio \(4 r ^ { 2 }\) is \(112 \frac { 1 } { 2 }\). Find the possible values of \(r\) and the corresponding values of \(a\). The function \(f ( x )\) is defined by $$f ( x ) = \frac { 6 x + 4 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) }$$ a) Express \(f ( x )\) in terms of partial fractions.
b) Find \(\int \frac { 3 x + 2 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) } \mathrm { d } x\), giving your answer in the form \(a \ln | g ( x ) |\), where \(a\) is a real number and \(g ( x )\) is a function of \(x\). Geraint opens a savings account. He deposits \(\pounds 10\) in the first month. In each subsequent month, the amount he deposits is 20 pence greater than the amount he deposited in the previous month.
a) Find the amount that Geraint deposits into the savings account in the 12th month.
b) Determine the number of months it takes for the total amount in the savings account to reach \(\pounds 954\).
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0
The diagram below shows a sketch of the curves \(y = x ^ { 2 }\) and \(y = 8 \sqrt { x }\). \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-3_508_869_2094_623} Find the area of the region bounded by the two curves.

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]