1.04e Sequences: nth term and recurrence relations

6 A sequence is defined by $$a _ { n + 1 } = 2 a _ { n } + 3 a _ { n - 1 } \quad \text { with } a _ { 1 } = 1 \text { and } a _ { 2 } = 1 .$$ Using the method on page 5, show that the value to which the ratio of successive terms converges is 3 .
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1. A sequence \(a _ { 1 } , a _ { 2 } a _ { 3 } , \ldots\) is defined by

$$a _ { n + 1 } = 5 - p a _ { n } \quad n \geq 1$$ where \(p\) is a constant.
Given that

• \(a _ { 1 } = 4\)
• the sequence is a periodic sequence of order 2.
  a. Write down an expression for \(a _ { 2 }\) and \(a _ { 3 }\).
  b. Find the value of \(p\).
  c. Find \(\sum _ { r = 1 } ^ { 21 } a _ { r }\)

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

$$a _ { n + 1 } = \frac { k \left( a _ { n } + 2 \right) } { a _ { n } } \quad n \in \mathbb { N }$$ where \(k\) is a constant.
Given that

• the sequence is a periodic sequence of order 3
• \(a _ { 1 } = 2\)
  1. show that

$$k ^ { 2 } + k - 2 = 0$$

For this sequence explain why \(k \neq 1\)

Find the value of $$\sum _ { r = 1 } ^ { 80 } a _ { r }$$

1. Given that

$$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 5 \quad x \in \mathbb { R }$$

1. express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers to be found. The curve with equation \(y = \mathrm { f } ( x )\)
   • meets the \(y\)-axis at the point \(P\)
   • has a minimum turning point at the point \(Q\)
   • Write down
     1. the coordinates of \(P\)
     2. the coordinates of \(Q\)

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

$$\begin{aligned} a _ { 1 } & = 3 \\ a _ { n + 1 } & = \frac { a _ { n } - 3 } { a _ { n } - 2 } , \quad n \in \mathbb { N } \end{aligned}$$

1. Find \(\sum _ { r = 1 } ^ { 100 } a _ { r }\)
2. Hence find \(\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }\)

1. (i) Show that \(\sum _ { r = 1 } ^ { 16 } \left( 3 + 5 r + 2 ^ { r } \right) = 131798\) (ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by

$$u _ { n + 1 } = \frac { 1 } { u _ { n } } , \quad u _ { 1 } = \frac { 2 } { 3 }$$ Find the exact value of \(\sum _ { r = 1 } ^ { 100 } u _ { r }\)

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

$$\begin{aligned} a _ { 1 } & = 3 \\ a _ { n + 1 } & = 8 - a _ { n } \end{aligned}$$

1. 1. Show that this sequence is periodic.
   2. State the order of this periodic sequence.
2. Find the value of $$\sum _ { n = 1 } ^ { 85 } a _ { n }$$

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

$$\begin{aligned} u _ { 1 } & = 35 \\ u _ { n + 1 } & = u _ { n } + 7 \cos \left( \frac { n \pi } { 2 } \right) - 5 ( - 1 ) ^ { n } \end{aligned}$$

1. 1. Show that \(u _ { 2 } = 40\)
   2. Find the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\) Given that the sequence is periodic with order 4
2. 1. write down the value of \(u _ { 5 }\)
   2. find the value of \(\sum _ { r = 1 } ^ { 25 } u _ { r }\)

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

$$\begin{aligned} u _ { n + 1 } & = k u _ { n } - 5 \\ u _ { 1 } & = 6 \end{aligned}$$ where \(k\) is a positive constant.
Given that \(u _ { 3 } = - 1\)

1. show that $$6 k ^ { 2 } - 5 k - 4 = 0$$
2. Hence
   1. find the value of \(k\),
   2. find the value of \(\sum _ { r = 1 } ^ { 3 } u _ { r }\)

7 A sequence is defined by the recurrence relation \(\mathrm { u } _ { \mathrm { k } + 1 } = \mathrm { u } _ { \mathrm { k } } + 5\) with \(\mathrm { u } _ { 1 } = - 2\).

1. Write down the values of \(u _ { 2 } , u _ { 3 }\), and \(u _ { 4 }\).
2. Explain whether this sequence is divergent or convergent.
3. Determine the value of \(u _ { 30 }\).
4. Determine the value of \(\sum _ { \mathrm { k } = 1 } ^ { 30 } \mathrm { u } _ { \mathrm { k } }\).

3 An infinite sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by \(a _ { \mathrm { n } } = \frac { \mathrm { n } } { \mathrm { n } + 1 }\), for all positive integers \(n\).

1. Find the limit of the sequence.
2. Prove that this is an increasing sequence.

11 The curve \(y = \mathrm { f } ( x )\) is defined by the function \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\) with domain \(0 \leq x \leq 4 \pi\).

1. 1. Show that the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\), when arranged in increasing order, form an arithmetic sequence.
   2. Show that the corresponding \(y\)-coordinates form a geometric sequence.
2. Would the result still hold with a larger domain? Give reasons for your answer.

7 Binet's formula for the \(n\)th Fibonacci number is given by \(\mathrm { F } _ { \mathrm { n } } = \frac { 1 } { \sqrt { 5 } } \left( \alpha ^ { \mathrm { n } } - \beta ^ { \mathrm { n } } \right)\) for \(n \geqslant 0\), where \(\alpha\) and \(\beta\) (with \(\alpha > 0 > \beta\) ) are the roots of \(x ^ { 2 } - x - 1 = 0\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Consider the sequence \(\left\{ \mathrm { S } _ { \mathrm { n } } \right\}\), where \(\mathrm { S } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } }\) for \(n \geqslant 0\).
   1. Determine the values of \(S _ { 2 }\) and \(S _ { 3 }\).
   2. Show that \(S _ { n + 2 } = S _ { n + 1 } + S _ { n }\) for \(n \geqslant 0\).
   3. Deduce that \(S _ { n }\) is an integer for all \(n \geqslant 0\).
3. A student models the terms of the sequence \(\left\{ \mathrm { S } _ { \mathrm { n } } \right\}\) using the formula \(\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } }\).
   1. Explain why this formula is unsuitable for every \(n \geqslant 1\).
   2. Considering the cases \(n\) even and \(n\) odd separately, state a modification of the formula \(\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } }\), other than \(\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } }\), such that \(\mathrm { T } _ { \mathrm { n } } = \mathrm { S } _ { \mathrm { n } }\) for all \(n \geqslant 1\).

5 In a conservation project in a nature reserve, scientists are modelling the population of one species of animal. The initial population of the species, \(P _ { 0 }\), is 10000 . After \(n\) years, the population is \(P _ { n }\). The scientists believe that the year-on-year change in the population can be modelled by a recurrence relation of the form \(\mathrm { P } _ { \mathrm { n } + 1 } = 2 \mathrm { P } _ { \mathrm { n } } \left( 1 - \mathrm { k } \mathrm { P } _ { \mathrm { n } } \right)\) for \(n \geqslant 0\), where \(k\) is a constant.

1. The initial aim of the project is to ensure that the population remains constant. Show that this happens, according to this model, when \(k = 0.00005\).
2. After a few years, with the population still at 10000 , the scientists suggest increasing the population. One way of achieving this is by adding 50 more of these animals into the nature reserve at the end of each year. In this scenario, the recurrence system modelling the population (using \(k = 0.00005\) ) is given by \(P _ { 0 } = 10000\) and \(\mathrm { P } _ { \mathrm { n } + 1 } = 2 \mathrm { P } _ { \mathrm { n } } \left( 1 - 0.00005 \mathrm { P } _ { \mathrm { n } } \right) + 50\) for \(n \geqslant 0\).
   Use your calculator to find the long-term behaviour of \(P _ { n }\) predicted by this recurrence system.
3. However, the scientists decide not to add any animals at the end of each year. Also, further research predicts that certain factors will remove 2400 animals from the population each year.
   1. Write down a modified form of the recurrence relation given in part (b), that will model the population of these animals in the nature reserve when 2400 animals are removed each year and no additional animals are added.
   2. Use your calculator to find the behaviour of \(P _ { n }\) predicted by this modified form of the recurrence relation over the course of the next ten years.
   3. Show algebraically that this modified form of the recurrence relation also gives a constant value of \(P _ { n }\) in the long term, which should be stated.
   4. Determine what constant value should replace 0.00005 in this modified form of the recurrence relation to ensure that the value of \(P _ { n }\) remains constant at 10000 .

1. A sequence is defined by the recurrence relation

$$u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , \quad n = 1,2,3 , \ldots ,$$ 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.
2. Given instead that \(u _ { 1 } = u _ { 2 } = 3\),
   1. calculate the value of \(a\),
   2. write down the value of \(u _ { 5 }\).

7. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation $$u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000 .$$ 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. Given that \(d = 100000\),
2. show that the population of fish dies out during the sixth year.
3. Find the value of \(d\) which would leave the population each year unchanged.

6. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation \(\quad u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000\).
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. Given that \(d = 100000\),
2. show that the population of fish dies out during the sixth year.
3. Find the value of \(d\) which would leave the population each year unchanged.

2. A sequence is defined by the recurrence relation \(u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , n = 1,2,3 , \ldots\), 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.
2. Given instead that \(u _ { 1 } = u _ { 2 } = 3\),
   1. calculate the value of \(a\),
   2. write down the value of \(u _ { 5 }\).

3. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 2 ^ { n } + k n ,$$ where \(k\) is a constant. Given that \(u _ { 1 } = u _ { 3 }\),

1. find the value of \(k\),
2. find the value of \(u _ { 5 }\).

7. (a) An arithmetic series has a common difference of 7 . Given that the sum of the first 20 terms of the series is 530 , find

1. the first term of the series,
2. the smallest positive term of the series.
   (b) The terms of a sequence are given by $$u _ { n } = ( n + k ) ^ { 2 } , \quad n \geq 1 ,$$ where \(k\) is a positive constant.
   Given that \(u _ { 2 } = 2 u _ { 1 }\),
3. find the value of \(k\),
4. show that \(u _ { 3 } = 11 + 6 \sqrt { 2 }\).

1. The \(n\)th term of a sequence is defined by

$$u _ { n } = n ^ { 2 } - 6 n + 11 , \quad n \geq 1 .$$ Given that the \(k\) th term of the sequence is 38 , find the value of \(k\).

1. A sequence of terms \(\left\{ t _ { n } \right\}\) is defined for \(n \geq 1\) by the recurrence relation

$$t _ { n + 1 } = k t _ { n } - 7 , \quad t _ { 1 } = 3$$ where \(k\) is a constant.

1. Find expressions for \(t _ { 2 }\) and \(t _ { 3 }\) in terms of \(k\). Given that \(t _ { 3 } = 13\),
2. find the possible values of \(k\).

5 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where \(p\) and \(q\) are constants. The first three terms of the sequence are given by $$u _ { 1 } = 200 \quad u _ { 2 } = 150 \quad u _ { 3 } = 120$$

1. Show that \(p = 0.6\) and find the value of \(q\).
2. Find the value of \(u _ { 4 }\).
3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\).

5 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where \(p\) and \(q\) are constants.
The first three terms of the sequence are given by $$u _ { 1 } = 200 \quad u _ { 2 } = 150 \quad u _ { 3 } = 120$$

1. Show that \(p = 0.6\) and find the value of \(q\).
2. Find the value of \(u _ { 4 }\).
3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\).

6 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where \(p\) and \(q\) are constants.
The first three terms of the sequence are given by $$u _ { 1 } = - 8 \quad u _ { 2 } = 8 \quad u _ { 3 } = 4$$

1. Show that \(q = 6\) and find the value of \(p\).
2. Find the value of \(u _ { 4 }\).
3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\).
   1. Write down an equation for \(L\).
   2. Hence find the value of \(L\).