Periodic Sequences

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 $$\sum _ { n = 1 } ^ { 70 } u _ { n }$$

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

$$a _ { n + 1 } = p + \frac { q } { a _ { n } } \quad n \in \mathbb { N }$$ where \(p\) and \(q\) are real numbers with \(q \neq 0\) It is known that

• one of the terms of this sequence is a
• the sequence is periodic
  1. Determine an equation for \(q\), in terms of \(p\) and \(a\), such that the sequence is constant (of period/order one).
  2. Determine the value of \(p\) that is necessary for the sequence to be of period/order 2.
  3. Give an example of a sequence that satisfies the condition in part (b), but is not of period/order 2.
  4. Determine an equation for \(q\), in terms of \(p\) only, such that the sequence has period/order 4.

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 $$\sum _ { r = 1 } ^ { 80 } a _ { r }$$

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.

2 For all positive integers \(n\), the terms of the sequence \(\left\{ u _ { n } \right\}\) are given by the formula \(u _ { n } = 3 n ^ { 2 } + 3 n + 7 ( \bmod 10 )\).

1. Show that \(u _ { n + 5 } = u _ { n }\) for all positive integers \(n\).
2. Hence describe the behaviour of the sequence, justifying your answer.

1 The sequence \(\left\{ u _ { n } \right\}\) is defined by \(u _ { 0 } = 2 , u _ { 1 } = 5\) and \(u _ { n } = \frac { 1 + u _ { n - 1 } } { u _ { n - 2 } }\) for \(n \geqslant 2\).
Prove that the sequence is periodic with period 5.

9 For each value of \(k\) the sequence of real numbers \(\left\{ u _ { n } \right\}\) is given by \(u _ { 1 } = 2\) and \(u _ { n + 1 } = \frac { k } { 6 + u _ { n } }\). For each of the following cases, either determine a value of \(k\) or prove that one does not exist.

1. \(\left\{ \mathrm { u } _ { n } \right\}\) is constant.
2. \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is periodic, with period 2 .
3. \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is periodic, with period 4 .

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.

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 is defined by $$u_1 = 3$$ $$u_{n+1} = 2 - \frac{4}{u_n}, \quad n \geq 1$$ Find the exact values of

1. \(u_2\), \(u_3\) and \(u_4\) [3]
2. \(u_{61}\) [1]
3. \(\sum_{i=1}^{99} u_i\) [3]

A sequence of numbers \(a_1, a_2, a_3, \ldots\) is defined by $$a_{n+1} = \frac{k(a_n + 2)}{a_n}$$, \(n \in \mathbb{N}\) where \(k\) is a constant. Given that

• the sequence is a periodic sequence of order 3
• \(a_1 = 2\)

1. show that $$k^2 + k - 2 = 0$$ [3]
2. For this sequence explain why \(k \neq 1\) [1]
3. Find the value of $$\sum_{r=1}^{80} a_r$$ [3]