1.04e Sequences: nth term and recurrence relations

9. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , u _ { 4 } , \ldots\) is defined by $$u _ { n + 1 } = 4 u _ { n } + 2 , \quad u _ { 1 } = 2$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = \frac { 2 } { 3 } \left( 4 ^ { n } - 1 \right)$$

7. A sequence can be described by the recurrence formula $$u _ { n + 1 } = 2 u _ { n } + 1 , \quad n \geqslant 1 , \quad u _ { 1 } = 1$$

1. Find \(u _ { 2 }\) and \(u _ { 3 }\).
2. Prove by induction that \(u _ { n } = 2 ^ { n } - 1\)

A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{array} { l l } a _ { n + 1 } = 4 a _ { n } - 3 , & n \geqslant 1 \\ a _ { 1 } = k , & \text { where } k \text { is a positive integer. } \end{array}$$

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\). Given that \(\sum _ { r = 1 } ^ { 3 } a _ { r } = 66\)
2. find the value of \(k\).

A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$a _ { 1 } = k , \quad a _ { n + 1 } = 4 a _ { n } - 7 ,$$ where \(k\) is a constant.

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Find \(a _ { 3 }\) in terms of \(k\), simplifying your answer. Given that \(a _ { 3 } = 13\),
3. find the value of \(k\).

2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 \quad \text { and } \quad u _ { n + 1 } = \frac { 1 } { 1 - u _ { n } } \text { for } n \geqslant 1 .$$

1. Write down the values of \(u _ { 2 } , u _ { 3 } , u _ { 4 }\) and \(u _ { 5 }\).
2. Deduce the value of \(u _ { 200 }\), showing your reasoning.

2 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 \quad \text { and } \quad u _ { n + 1 } = 1 - u _ { n } \text { for } n \geqslant 1 .$$

1. Write down the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
2. Find \(\sum _ { n = 1 } ^ { 100 } u _ { n }\).

6 A sequence is given by $$\begin{gathered} a _ { 1 } = 4 \\ a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.

4 Sequences \(\mathrm { A } , \mathrm { B }\) and C are shown below. They each continue in the pattern established by the given terms. $$\begin{array} { l l l l l l l l l } \text { A: } & 1 , & 2 , & 4 , & 16 , & 32 , & \ldots & \\ \text { B: } & 20 , & - 10 , & 5 , & - 2.5 , & 1.25 , & - 0.625 , & \ldots \\ \text { C: } & 20 , & 5 , & 1 , & 20 , & 5 , & 1 , & \ldots \end{array}$$

1. Which of these sequences is periodic?
2. Which of these sequences is convergent?
3. Find, in terms of \(n\), the \(n\)th term of sequence A .

2 A sequence begins $$\begin{array} { l l l l l l l l l l l l } 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 & 1 & \ldots \end{array}$$ and continues in this pattern.

1. Find the 48th term of this sequence.
2. Find the sum of the first 48 terms of this sequence.

4

1. Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5 \\ t _ { 1 } & = 3 \end{aligned}$$
2. Find \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).

5 A sequence is defined by \(a _ { k } = 5 k + 1\), for \(k = 1,2,3 \ldots\)

1. Write down the first three terms of the sequence.
2. Evaluate \(\sum _ { k = 1 } ^ { 100 } a _ { k }\).

11 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \ldots\) where \(a _ { 1 }\) is a given real number is defined by \(a _ { n + 1 } = 1 - \frac { 1 } { a _ { n } }\).

1. For the case when \(a _ { 1 } = 2\), find \(a _ { 2 } , a _ { 3 }\) and \(a _ { 4 }\). Describe the behaviour of this sequence
2. For the case when \(a _ { 1 } = k\), where \(k\) is an integer greater than 1 , find \(a _ { 2 }\) in terms of \(k\) as a single fraction.
   Find also \(a _ { 3 }\) in its simplest form and hence deduce that \(a _ { 4 } = k\).
3. Show that \(a _ { 2 } a _ { 3 } a _ { 4 } = - 1\) for any integer \(k\).
4. When \(a _ { 1 } = 2\) evaluate \(\sum _ { i = 1 } ^ { 99 } a _ { i }\).

7 For each of the following sequences, write down sufficient terms of the sequence in order to be able to describe its behaviour as divergent, periodic or convergent. For any convergent sequence, state its limit.

1. \(a _ { 1 } = - 1 ; \quad a _ { k + 1 } = \frac { 4 } { a _ { k } }\)
2. \(\quad a _ { 1 } = 1 ; \quad a _ { k } = 2 - 2 \times \left( \frac { 1 } { 2 } \right) ^ { k }\)
3. \(\quad a _ { 1 } = 0 \quad a _ { k + 1 } = \left( 1 + a _ { k } \right) ^ { 2 }\).

1. A sequence is defined by

$$u _ { n + 1 } = \frac { u _ { n } + 1 } { 3 } , \quad n = 1,2,3 , \ldots$$ Given that \(u _ { 3 } = 5\),

1. find the value of \(u _ { 4 }\),
2. find the value of \(u _ { 1 }\).

1. 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. The same sequence can also be defined by the recurrence relation $$u _ { n + 1 } = a u _ { n } + b , \quad n \geq 1 , \quad u _ { 1 } = 1 ,$$ where \(a\) and \(b\) are constants.
2. Find the values of \(a\) and \(b\).

1. 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 \(u _ { 5 }\).

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

1. find expressions for \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\). Given also that \(u _ { 2 } + u _ { 3 } = 11\),
2. find the possible values of \(k\).

3 A sequence is given by $$\begin{gathered} a _ { 1 } = 4 \\ a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.

1 A sequence is defined by \(u _ { 1 } = 2\) and \(u _ { k + 1 } = \frac { 10 } { u _ { k } ^ { 2 } }\).
Calculate \(\sum _ { k = 1 } ^ { 4 } u _ { k }\).

2 The \(n\)th term of a sequence, \(u _ { n }\), is given by $$u _ { n } = 12 - \frac { 1 } { 2 } n .$$

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\). State what type of sequence this is.
2. Find \(\sum _ { n = 1 } ^ { 30 } u _ { n }\).

3 A sequence is defined by $$\begin{aligned} u _ { 1 } & = 10 \\ u _ { r + 1 } & = \frac { 5 } { u _ { r } ^ { 2 } } \end{aligned}$$ Calculate the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
What happens to the terms of the sequence as \(r\) tends to infinity?

4 The \(n\)th term, \(t _ { n }\), of a sequence is given by $$t _ { n } = \sin ( \theta + 180 n ) ^ { \circ }$$ Express \(t _ { 1 }\) and \(t _ { 2 }\) in terms of \(\sin \theta ^ { \circ }\).

6 You are given that $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = \frac { u _ { n } } { 1 + u _ { n } } \end{aligned}$$ Find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\). Give your answers as fractions.

9

1. Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5 \\ t _ { 1 } & = 3 \end{aligned}$$
2. Find \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).