Recurrence relation evaluation

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 \(k\).

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 \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).

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 \(b\).

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 \(k\).

6 Find the second and third terms in the sequence given by $$\begin{aligned} & u _ { 1 } = 5 \\ & u _ { n + 1 } = u _ { n } + 3 \end{aligned}$$ Find also the sum of the first 50 terms of this sequence.

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 \(\sum _ { k = 1 } ^ { 4 } u _ { k }\).

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?

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 \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).

5. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\) is defined by $$a _ { n + 1 } = 5 a _ { n } - 3 , \quad n \geqslant 1$$ Given that \(a _ { 2 } = 7\),

1. find the value of \(a _ { 1 }\)
2. Find the value of \(\sum _ { r = 1 } ^ { 4 } a _ { r }\)

8 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 8 \quad \text { and } \quad u _ { n + 1 } = u _ { n } + 3 .$$

1. Show that \(u _ { 5 } = 20\).
2. The \(n\)th term of the sequence can be written in the form \(u _ { n } = p n + q\). State the values of \(p\) and \(q\).
3. State what type of sequence it is.
4. Find the value of \(N\) such that \(\sum _ { n = 1 } ^ { 2 N } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } = 1256\).

2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 7 \text { and } u _ { n + 1 } = u _ { n } + 4 \text { for } n \geqslant 1 .$$

1. Show that \(u _ { 17 } = 71\).
2. Show that \(\sum _ { n = 1 } ^ { 35 } u _ { n } = \sum _ { n = 36 } ^ { 50 } u _ { n }\).

5

1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 4 \quad \text { and } \quad u _ { n + 1 } = \frac { 2 } { u _ { n } } \quad \text { for } n \geqslant 1 .$$
   1. Write down the values of \(u _ { 2 }\) and \(u _ { 3 }\).
   2. Describe the behaviour of the sequence.
2. In an arithmetic progression the ninth term is 18 and the sum of the first nine terms is 72. Find the first term and the common difference.

2 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?

2 Find the second and third terms in the sequence given by $$\begin{aligned} & u _ { 1 } = 5 \\ & u _ { n + 1 } = u _ { n } + 3 . \end{aligned}$$ Find also the sum of the first 50 terms of this sequence.

2 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 A sequence is defined as follows. \(u _ { 1 } = a\), where \(a > 0\) To obtain \(u _ { r + 1 }\)

• find the remainder when \(u _ { r }\) is divided by 3 ,
• multiply the remainder by 5 ,
• the result is \(u _ { r + 1 }\).

Find \(\sum _ { r = 2 } ^ { 4 } u _ { r }\) in each of the following cases.

1. \(a = 5\)
2. \(a = 6\)

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

$$u _ { n + 1 } = k - \frac { 24 } { u _ { n } } \quad u _ { 1 } = 2$$ where \(k\) is an integer.
Given that \(u _ { 1 } + 2 u _ { 2 } + u _ { 3 } = 0\)

1. show that $$3 k ^ { 2 } - 58 k + 240 = 0$$
2. Find the value of \(k\), giving a reason for your answer.
3. Find the value of \(u _ { 3 }\)

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. Jamie takes out an interest-free loan of \(\pounds 8100\)

Jamie makes a payment every month to pay back the loan.
Jamie repays \(\pounds 400\) in month \(1 , \pounds 390\) in month \(2 , \pounds 380\) in month 3 , and so on, so that the amounts repaid each month form an arithmetic sequence.

1. Show that Jamie repays \(\pounds 290\) in month 12 After Jamie's \(N\) th payment, the loan is completely paid back.
2. Show that \(N ^ { 2 } - 81 N + 1620 = 0\)
3. Hence find the value of \(N\).

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 \(\sum _ { r = 1 } ^ { 3 } u _ { r }\)