1.04f Sequence types: increasing, decreasing, periodic

1. 1. Find the value of

$$\sum _ { r = 1 } ^ { \infty } 6 \times ( 0.25 ) ^ { r }$$ (3)
(ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$\begin{aligned} u _ { 1 } & = 3 \\ u _ { n + 1 } & = \frac { u _ { n } - 3 } { u _ { n } - 2 } \quad n \in \mathbb { N } \end{aligned}$$

1. Show that this sequence is periodic.
2. State the order of this sequence.
3. Hence find $$\sum _ { n = 1 } ^ { 70 } u _ { n }$$

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.

3

1. Find \(\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)\).
2. State whether the sequence with \(k\) th term \(k ^ { 2 } - 1\) is convergent or divergent, giving a reason for your answer.

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. \(\quad a _ { 1 } = 0 \quad a _ { k + 1 } = \left( 1 + a _ { k } \right) ^ { 2 }\).

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?

8

1. Find \(\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)\).
2. State whether the sequence with \(k\) th term \(k ^ { 2 } - 1\) is convergent or divergent, giving a reason for your answer.

10 For each of the following sequences, state with a reason whether it is convergent, periodic or neither. Each sequence continues in the pattern established by the given terms.

1. \(3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \ldots\)
2. \(3,7,11,15 , \ldots\)
3. \(3,5 , - 3 , - 5,3,5 , - 3 , - 5 , \ldots\)

3 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 Sequences A, B and C are shown below. They each continue in the pattern established by the given terms.

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 .

4. (a) Show that \(\sum _ { r = 1 } ^ { 20 } \left( 2 ^ { r - 1 } - 3 - 4 r \right) = 1047675\) (b) A sequence has \(n\)th term \(u _ { n } = \sin \left( 90 n ^ { \circ } \right) n \geq 1\)

1. Find the order of the sequence.
2. Find \(\sum _ { r = 1 } ^ { 222 } u _ { r }\)

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 \(\sum _ { r = 1 } ^ { 25 } u _ { r }\)

4

1. The first four terms of a sequence are \(2,3,0,3\) and the subsequent terms are given by \(\mathrm { a } _ { \mathrm { k } + 4 } = \mathrm { a } _ { \mathrm { k } }\).
   1. State what type of sequence this is.
   2. Find \(\sum _ { \mathrm { k } = 1 } ^ { 200 } \mathrm { a } _ { \mathrm { k } }\).
2. A different sequence is given by \(\mathrm { u } _ { \mathrm { n } } = \mathrm { b } ^ { \mathrm { n } }\) where \(b\) is a constant and \(n \geqslant 1\).
   1. State the set of values of \(b\) for which this is a divergent sequence.
   2. In the case where \(b = \frac { 1 } { 3 }\), find the sum of all the terms in the sequence.

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 \(\sum _ { \mathrm { k } = 1 } ^ { 30 } \mathrm { u } _ { \mathrm { k } }\).

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.

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 second term of the sequence is 160 . The third term of the sequence is 132 .
The limit of \(u _ { n }\) as \(n\) tends to infinity is 20 .

1. Find the value of \(p\) and the value of \(q\).
2. Hence find the value of the first term of the sequence.

3 A periodic sequence is defined by \(U _ { n } = \sin \left( \frac { n \pi } { 2 } \right)\) State the period of this sequence. Circle your answer. \(82 \pi \quad 4 \quad \pi\)

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 \(L\)

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 \(n\)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) \(u _ { 6 } = 33 , u _ { n } = 2 u _ { n - 1 } - 1\)

A sequence begins $$1 \quad 3 \quad 5 \quad 3 \quad 1 \quad 3 \quad 5 \quad 3 \quad 1 \quad 3 \quad \ldots$$ and continues in this pattern.

1. Find the 55th term of this sequence, showing your method. [1]
2. Find the sum of the first 55 terms of the sequence. [2]

For each of the following sequences, state with a reason whether it is convergent, periodic or neither. Each sequence continues in the pattern established by the given terms.

1. \(3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \ldots\) [1]
2. \(3, 7, 11, 15, \ldots\) [1]
3. \(3, 5, -3, -5, 3, 5, -3, -5, \ldots\) [1]