Recurrence relation: evaluate sum

1. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies

$$u _ { n + 1 } = 2 u _ { n } - 6 , \quad n \geqslant 1$$ Given that \(u _ { 1 } = 2\)

1. find the value of \(u _ { 3 }\)
2. evaluate \(\sum _ { i = 1 } ^ { 4 } u _ { i }\)

2. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = 2 - 3 u _ { n } \quad n \geqslant 1 \end{aligned}$$

1. Find the value of \(u _ { 2 }\) and the value of \(u _ { 3 }\)
2. Calculate the value of \(\sum _ { r = 1 } ^ { 4 } \left( r - u _ { r } \right)\) □

5. (i) $$U _ { n + 1 } = \frac { U _ { n } } { U _ { n } - 3 } , \quad n \geqslant 1$$ Given \(U _ { 1 } = 4\), find

1. \(U _ { 2 }\)
2. \(\sum _ { n = 1 } ^ { 100 } U _ { n }\) (ii) Given $$\sum _ { r = 1 } ^ { n } ( 100 - 3 r ) < 0$$ find the least value of the positive integer \(n\).

7. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3 \\ u _ { n + 1 } & = u _ { n } - 5 , \quad n \geqslant 1 \end{aligned}$$ Find the values of

1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. \(u _ { 100 }\)
3. \(\sum _ { i = 1 } ^ { 100 } u _ { i }\)

4. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies $$u _ { n + 1 } = 2 u _ { n } - 1 , n \geqslant 1$$ Given that \(u _ { 2 } = 9\),

1. find the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\),
2. evaluate \(\sum _ { r = 1 } ^ { 4 } u _ { r }\).

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

1. Find the value of \(a _ { 2 }\) and the value of \(a _ { 3 }\).
2. Calculate the value of \(\sum _ { r = 1 } ^ { 5 } a _ { r }\).

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

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Show that \(a _ { 3 } = 9 k + 20\).
3. 1. Find \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) in terms of \(k\).
   2. Show that \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) is divisible by 10 .

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

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Show that \(a _ { 3 } = 25 k + 18\).
3. 1. Find \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) in terms of \(k\), in its simplest form.
   2. Show that \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) is divisible by 6 .

6. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } \ldots\) is defined by $$\begin{gathered} x _ { 1 } = 1 \\ x _ { n + 1 } = \left( x _ { n } \right) ^ { 2 } - k x _ { n } , \quad n \geqslant 1 \end{gathered}$$ where \(k\) is a constant, \(k \neq 0\)

1. Find an expression for \(x _ { 2 }\) in terms of \(k\).
2. Show that \(x _ { 3 } = 1 - 3 k + 2 k ^ { 2 }\) Given also that \(x _ { 3 } = 1\),
3. calculate the value of \(k\).
4. Hence find the value of \(\sum _ { n = 1 } ^ { 100 } x _ { n }\)

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

1. Write down expressions for \(a _ { 2 }\) and \(a _ { 3 }\) in terms of \(k\). Find
2. \(\sum _ { r = 1 } ^ { 3 } \left( 1 + a _ { r } \right)\) in terms of \(k\), giving your answer in its simplest form,
3. \(\sum _ { r = 1 } ^ { 100 } \left( a _ { r + 1 } + k a _ { r } \right)\)

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

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 \(\sum _ { r = 1 } ^ { 4 } a _ { r }\)

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

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 \(\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }\)

3 A sequence is defined by $$u _ { 1 } = a \text { and } u _ { n + 1 } = - 1 \times u _ { n }$$ Find \(\sum _ { n = 1 } ^ { 95 } u _ { n }\) Circle your answer. \(- a\) 0 \(a\) 95a

A sequence is defined by the recurrence relation $$u_{n+1} = u_n - 2, \quad n > 0, \quad u_1 = 50.$$

1. Write down the first four terms of the sequence. [1]
2. Evaluate $$\sum_{r=1}^{20} u_r.$$ [3]