1.04e Sequences: nth term and recurrence relations

2 A periodic sequence is defined by $$U _ { n } = ( - 1 ) ^ { n }$$ State the period of the sequence.
Circle your answer.

11 The \(n\)th term of a sequence is \(u _ { n }\) The sequence is defined by $$u _ { n + 1 } = p u _ { n } + 70$$ where \(u _ { 1 } = 400\) and \(p\) is a constant.
11

1. Find an expression, in terms of \(p\), for \(u _ { 2 }\) 11
2. It is given that \(u _ { 3 } = 382\) 11 (b) (i) Show that \(p\) satisfies the equation $$200 p ^ { 2 } + 35 p - 156 = 0$$ 11 (b) (ii) It is given that the sequence is a decreasing sequence. Find the value of \(u _ { 4 }\) and the value of \(u _ { 5 }\) 11
3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\) 11 (c) (i) Write down an equation for \(L\) 11 (c) (ii) Find the value of \(L\)

3 A sequence is defined by $$u _ { 1 } = a \text { and } u _ { n + 1 } = - 1 \times u _ { n }$$ Find \(\sum _ { n = 1 } ^ { 95 } u _ { n }\) Circle your answer. \(- a\) 0 \(a\) 95a

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. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by

$$\begin{gathered} u _ { 1 } = 7 \\ u _ { n + 1 } = ( - 1 ) ^ { n } u _ { n } + k \end{gathered}$$ where \(k\) is a constant.

1. Show that \(u _ { 5 } = 7\) Given that \(\sum _ { r = 1 } ^ { 4 } u _ { r } = 30\)
2. find the value of \(k\).
3. Hence find the value of \(\sum _ { r = 1 } ^ { 150 } u _ { r }\)

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.

9 A new lake is stocked with fish. Let \(P _ { t }\) be the population of fish in the lake after \(t\) years. Two models using recurrence relations are proposed for \(P _ { t }\), with \(P _ { 0 } = 550\). $$\begin{aligned} & \text { Model } 1 : P _ { t } = 2 P _ { t - 1 } \mathrm { e } ^ { - 0.001 P _ { t - 1 } } \\ & \text { Model } 2 : P _ { t } = \frac { 1 } { 2 } P _ { t - 1 } \left( 7 - \frac { 1 } { 160 } P _ { t - 1 } \right) \end{aligned}$$

1. Evaluate the population predicted by each model when \(t = 3\).
2. Identify, with evidence, which one of the models predicts a stable population in the long term.
3. Describe the long term behaviour of the population for the other model.

4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), is defined by \(u _ { n } = 3 n + 5\).

1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find the value of \(n\) such that \(u _ { n } = 254\).
3. Evaluate \(\sum _ { n = 1 } ^ { 500 } u _ { n }\).

11 A sequence of terms \(x _ { n }\) generated by a recurrence relation is said to be strictly increasing if, for each \(x _ { n } , x _ { n + 1 } > x _ { n }\).

1. Let a recurrence relation be defined by $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 } \quad \text { and } \quad x _ { 0 } = \frac { 1 } { 2 } \quad \text { for } n \geq 0$$ Calculate \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) correct to 3 significant figures where appropriate.
2. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 }$$ show that the sequence is strictly increasing when \(x _ { n } > 2\) or \(x _ { n } < 1\).
3. If \(- 1 < x _ { 0 } < 1\), then the sequence \(x _ { n } ( n \geq 0 )\) converges to a limit. Explain briefly why this limit is 1 .
4. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + k } { m } \text { with } m > 0$$ prove that \(x _ { n }\) is a strictly increasing sequence for all \(x _ { n }\) if \(m ^ { 2 } < 4 k\).

The \(n\)th term of a number sequence is denoted by \(x _ { n }\). The \(( n + 1 )\) th term is defined by \(x _ { n + 1 } = 4 x _ { n } - 3\) and \(x _ { 3 } = 113\). a) Find the values of \(x _ { 2 }\) and \(x _ { 1 }\).
b) Determine whether the sequence is an arithmetic sequence, a geometric sequence or neither. Give reasons for your answer.
a) Express \(5 \sin x - 12 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
b) Find the minimum value of \(\frac { 4 } { 5 \sin x - 12 \cos x + 15 }\).
c) Solve the equation $$5 \sin x - 12 \cos x + 3 = 0$$ for values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). a) Find the range of values of \(x\) for which \(| 1 - 3 x | > 7\).
b) Sketch the graph of \(y = | 1 - 3 x | - 7\). Clearly label the minimum point and the points where the graph crosses the \(x\)-axis.

For each of the following sequences, find the first 5 terms, \(u _ { 1 }\) to \(u _ { 5 }\). Describe the behaviour of each sequence. a) \(\quad u _ { n } = \sin \left( \frac { n \pi } { 2 } \right)\) b) \(u _ { 6 } = 33 , u _ { n } = 2 u _ { n - 1 } - 1\)

The sequence of positive numbers \(u_1, u_2, u_3, \ldots\) is given by $$u_{n+1} = (u_n - 3)^2, \quad u_1 = 1.$$

1. Find \(u_2\), \(u_3\) and \(u_4\). [3]
2. Write down the value of \(u_{20}\). [1]

A sequence \(a_1, a_2, a_3, \ldots\) is defined by $$a_1 = 3,$$ $$a_{n+1} = 3a_n - 5, \quad n \geq 1.$$

1. Find the value \(a_2\) and the value of \(a_3\). [2]
2. Calculate the value of \(\sum_{r=1}^5 a_r\). [3]

Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns: Row 1 \(\square\) Row 2 \(\square\square\) Row 3 \(\square\square\square\) She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make 2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.

1. Find an expression, in terms of \(n\), for the number of sticks required to make a similar arrangement of \(n\) squares in the \(n\)th row. [3]

Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.

1. Find the total number of sticks Ann uses in making these 10 rows. [3]

Ann started with 1750 sticks. Given that Ann continues the pattern to complete \(k\) rows but does not have sufficient sticks to complete the \((k + 1)\)th row,

1. show that \(k\) satisfies \((3k - 100)(k + 35) < 0\). [4]
2. Find the value of \(k\). [2]

Initially the number of fish in a lake is 500 000. The population is then modelled by the recurrence relation $$u_{n+1} = 1.05u_n - d, \quad u_0 = 500000.$$ In this relation \(u_n\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year. Given that \(d = 15000\),

1. calculate \(u_1\), \(u_2\) and \(u_3\) and comment briefly on your results. [3]

Given that \(d = 100000\),

1. show that the population of fish dies out during the sixth year. [3]
2. Find the value of \(d\) which would leave the population each year unchanged. [2]

A sequence is defined by the recurrence relation $$u_{n+1} = \sqrt{\frac{u_n}{2} + \frac{a}{u_n}}, \quad n = 1, 2, 3, \ldots,$$ where \(a\) is a constant.

1. Given that \(a = 20\) and \(u_1 = 3\), find the values of \(u_2\), \(u_3\) and \(u_4\), giving your answers to 2 decimal places. [3]
2. Given instead that \(u_1 = u_2 = 3\),
   1. calculate the value of \(a\), [3]
   2. write down the value of \(u_5\). [1]

The sequence \(u_1, u_2, u_3, \ldots, u_n\) is defined by the recurrence relation $$u_{n+1} = pu_n + 5, \quad u_1 = 2, \text{ where } p \text{ is a constant.}$$ Given that \(u_3 = 8\),

1. show that one possible value of \(p\) is \(\frac{1}{2}\) and find the other value of \(p\). [5]

Using \(p = \frac{1}{2}\),

1. write down the value of \(\log_2 p\). [1]

Given also that \(\log_2 q = t\),

1. express \(\log_2 \left(\frac{p^3}{\sqrt{q}}\right)\) in terms of \(t\). [3]

Given that for all positive integers \(n\), $$\sum_{r=1}^{n} a_r = 12 + 4n^2$$

1. find the value of \(\sum_{r=1}^{5} a_r\) [2]
2. Find the value of \(a_6\) [3]

The sequence \(u_1, u_2, u_3, ...\) is defined by the recurrence relation $$u_{n+1} = (u_n)^2 - 1, \quad n \geq 1.$$ Given that \(u_1 = k\), where \(k\) is a constant,

1. find expressions for \(u_2\) and \(u_3\) in terms of \(k\). [3]

Given also that \(u_2 + u_3 = 11\),

1. find the possible values of \(k\). [4]

A sequence is defined by the recurrence relation $$u_{n+1} = u_n - 2, \quad n > 0, \quad u_1 = 50.$$

1. Write down the first four terms of the sequence. [1]
2. Evaluate $$\sum_{r=1}^{20} u_r.$$ [3]

A sequence of terms is defined by $$u_n = 3^n - 2, \quad n \geq 1.$$

1. Write down the first four terms of the sequence. [2]

The same sequence can also be defined by the recurrence relation $$u_{n+1} = au_n + b, \quad n \geq 1, \quad u_1 = 1,$$ where \(a\) and \(b\) are constants.

1. Find the values of \(a\) and \(b\). [4]

A sequence is defined by the recurrence relation $$u_{n+1} = \frac{u_n + 1}{3}, \quad n = 1, 2, 3, ...$$ Given that \(u_3 = 5\),

1. find the value of \(u_4\), [1]
2. find the value of \(u_1\). [3]

The \(n\)th term of a sequence is \(u_n\). The sequence is defined by $$u_{n+1} = ku_n + 12$$ where \(k\) is a constant. The first two terms of the sequence are given by $$u_1 = 16 \quad u_2 = 24$$

1. Show that \(k = 0.75\). [2]
2. Find the value of \(u_3\) and the value of \(u_4\). [2]
3. The limit of \(u_n\) as \(n\) tends to infinity is \(L\).
   1. Write down an equation for \(L\). [1]
   2. Hence find the value of \(L\). [2]

The sequence \(u_1, u_2, u_3, \ldots, u_n\) is defined by the recurrence relation $$u_{n+1} = pu_n + 5, u_1 = 2, \text{ where } p \text{ is a constant.}$$ Given that \(u_3 = 8\),

1. show that one possible value of \(p\) is \(\frac{1}{2}\) and find the other value of \(p\). [5]

Using \(p = \frac{1}{2}\),

1. write down the value of \(\log_2 p\). [1]

Given also that \(\log_2 q = t\),

1. express \(\log_2 \left(\frac{p^3}{\sqrt{q}}\right)\) in terms of \(t\). [3]

You are given that $$u_1 = 1,$$ $$u_{n+1} = \frac{u_n}{1 + u_n}.$$ Find the values of \(u_2\), \(u_3\) and \(u_4\). Give your answers as fractions. [2]