Recurrence relation: find parameter from given term

1. A sequence is defined by

$$\begin{aligned} u _ { 1 } & = k \\ u _ { n + 1 } & = 3 u _ { n } - 12 , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.

1. Write down fully simplified expressions for \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\) in terms of \(k\). Given that \(u _ { 4 } = 15\)
2. find the value of \(k\),
3. find \(\sum _ { i = 1 } ^ { 4 } u _ { i }\), giving an exact numerical answer.

4. A sequence is defined by $$\begin{aligned} u _ { 1 } & = k , \text { where } k \text { is a constant } \\ u _ { n + 1 } & = 4 u _ { n } - 3 , n \geqslant 1 \end{aligned}$$

1. Find \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\), simplifying your answers as appropriate. Given \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 18\)
2. find \(k\).

5. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = k - \frac { 8 } { u _ { n } } \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.

1. Write down expressions for \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\). Given that \(u _ { 3 } = 6\)
2. find the possible values of \(k\).

7. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{gathered} a _ { 1 } = 2 \\ a _ { n + 1 } = 3 a _ { n } - c \end{gathered}$$ where \(c\) is a constant.

1. Find an expression for \(a _ { 2 }\) in terms of \(c\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 0\)
2. find the value of \(c\).

1. A sequence is given by:

$$\begin{aligned} & x _ { 1 } = 1 \\ & x _ { n + 1 } = x _ { n } \left( p + x _ { n } \right) \end{aligned}$$ where \(p\) is a constant ( \(p \neq 0\) ) .

1. Find \(x _ { 2 }\) in terms of \(p\).
2. Show that \(x _ { 3 } = 1 + 3 p + 2 p ^ { 2 }\). Given that \(x _ { 3 } = 1\),
3. find the value of \(p\),
4. write down the value of \(x _ { 2008 }\).

4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 2 \\ a _ { n + 1 } & = 3 a _ { n } - c \end{aligned}$$ where \(c\) is a constant.

1. Find an expression for \(a _ { 2 }\) in terms of \(c\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 0\)
2. find the value of \(c\).

4. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is defined by $$\begin{aligned} x _ { 1 } & = 1 \\ x _ { n + 1 } & = a x _ { n } + 5 , \quad n \geqslant 1 \end{aligned}$$ where \(a\) is a constant.

1. Write down an expression for \(x _ { 2 }\) in terms of \(a\).
2. Show that \(x _ { 3 } = a ^ { 2 } + 5 a + 5\) Given that \(x _ { 3 } = 41\)
3. find the possible values of \(a\).

5. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is defined by $$\begin{gathered} x _ { 1 } = 1 , \\ x _ { n + 1 } = a x _ { n } - 3 , n \geqslant 1 , \end{gathered}$$ where \(a\) is a constant.

1. Find an expression for \(x _ { 2 }\) in terms of \(a\).
2. Show that \(x _ { 3 } = a ^ { 2 } - 3 a - 3\). Given that \(x _ { 3 } = 7\),
3. find the possible values of \(a\).

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

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Show that \(a _ { 3 } = 4 k - 21\). Given that \(\sum _ { r = 1 } ^ { 4 } a _ { r } = 43\),
3. find the value of \(k\).

5. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\) is defined by $$\begin{aligned} & a _ { 1 } = 3 \\ & a _ { n + 1 } = 2 a _ { n } - c \quad ( n \geqslant 1 ) \end{aligned}$$ where \(c\) is a constant.

1. Write down an expression, in terms of \(c\), for \(a _ { 2 }\)
2. Show that \(a _ { 3 } = 12 - 3 c\) Given that \(\sum _ { i = 1 } ^ { 4 } a _ { i } \geqslant 23\)
3. find the range of values of \(c\).

4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 4 \\ a _ { n + 1 } & = k \left( a _ { n } + 2 \right) , \quad \text { for } n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.

1. Find an expression for \(a _ { 2 }\) in terms of \(k\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 2\),
2. find the two possible values of \(k\).

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

$$\begin{aligned} a _ { 1 } & = 1 \\ a _ { n + 1 } & = \frac { k \left( a _ { n } + 1 \right) } { a _ { n } } , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a positive constant.

1. Write down expressions for \(a _ { 2 }\) and \(a _ { 3 }\) in terms of \(k\), giving your answers in their simplest form. Given that \(\sum _ { r = 1 } ^ { 3 } a _ { r } = 10\)
2. find an exact value for \(k\).

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 \(k\).

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 \(k\).

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 \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) 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 \(\sum _ { r = 1 } ^ { 3 } u _ { r }\)

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 \(u _ { 5 }\).

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 \(u _ { 5 }\).

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 \(k\).

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. [2]

The same sequence can also be defined by the recurrence relation $$u_{n+1} = au_n + b, \quad n \geq 1, \quad u_1 = 1,$$ where \(a\) and \(b\) are constants.

1. Find the values of \(a\) and \(b\). [4]

A sequence of terms \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 4$$ $$a_{n+1} = ka_n + 3$$ where \(k\) is a constant. Given that • \(\sum_{n=1}^{5} a_n = 12\) • all terms of the sequence are different find the value of \(k\) [4]