1.04e Sequences: nth term and recurrence relations

3 The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49 .

1. Find the first term of the progression and the common difference. The \(n\)th term of the progression is 46 .
2. Find the value of \(n\).

5 The sequence of values given by the iterative formula \(x _ { n + 1 } = \frac { 6 + 8 x _ { n } } { 8 + x _ { n } ^ { 2 } }\) with initial value \(x _ { 1 } = 2\) converges to \(\alpha\).

1. Use the iterative formula to find the value of \(\alpha\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
2. State an equation satisfied by \(\alpha\) and hence determine the exact value of \(\alpha\).

3 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) defined by $$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { 1 } { 2 } \sqrt [ 3 ] { } \left( x _ { n } ^ { 2 } + 6 \right)$$ converges to the value \(\alpha\).

1. Find the value of \(\alpha\) correct to 3 decimal places. Show your working, giving each calculated value of the sequence to 5 decimal places.
2. Find, in the form \(a x ^ { 3 } + b x ^ { 2 } + c = 0\), an equation of which \(\alpha\) is a root.

3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } > 4\) and, for \(n \geqslant 1\), $$u _ { n + 1 } = \frac { u _ { n } ^ { 2 } + u _ { n } + 12 } { 2 u _ { n } }$$

1. By considering \(\mathrm { u } _ { \mathrm { n } + 1 } - 4\), or otherwise, prove by mathematical induction that \(\mathrm { u } _ { \mathrm { n } } > 4\) for all positive integers \(n\).
2. Show that \(u _ { n + 1 } < u _ { n }\) for \(n \geqslant 1\).

5. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3 \\ u _ { n + 1 } & = 2 - \frac { 4 } { u _ { n } } , \quad n \geqslant 1 \end{aligned}$$ Find the exact values of

1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. \(u _ { 61 }\)
3. \(\sum _ { i = 1 } ^ { 99 } u _ { i }\)

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.

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 \(\sum _ { r = 1 } ^ { 4 } \left( r - u _ { r } \right)\) □

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 \(k\).

10. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 4 \\ u _ { n + 1 } & = \frac { 2 u _ { n } } { 3 } , \quad n \geqslant 1 \end{aligned}$$

1. Find the exact values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. Find the value of \(u _ { 20 }\), giving your answer to 3 significant figures.
3. Evaluate $$12 - \sum _ { i = 1 } ^ { 16 } u _ { i }$$ giving your answer to 3 significant figures.
4. Explain why \(\sum _ { i = 1 } ^ { N } u _ { i } < 12\) for all positive integer values of \(N\).

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. \(\sum _ { i = 1 } ^ { 100 } u _ { i }\)

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 \(k\).

6. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 36 \\ u _ { n + 1 } & = \frac { 2 } { 3 } u _ { n } , \quad n \geqslant 1 \end{aligned}$$

1. Find the exact simplified values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. Write down the common ratio of the sequence.
3. Find, giving your answer to 4 significant figures, the value of \(u _ { 11 }\)
4. Find the exact value of \(\sum _ { i = 1 } ^ { 6 } u _ { i }\)
5. Find the value of \(\sum _ { i = 1 } ^ { \infty } u _ { i }\)

4. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies $$u _ { n } = k n - 3 ^ { n }$$ where \(k\) is a constant. Given that \(u _ { 2 } = u _ { 4 }\)

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

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 \(c\).

2. The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by: $$u _ { n + 1 } = \left( u _ { n } - 3 \right) ^ { 2 } , \quad u _ { 1 } = 1 .$$

1. Find \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
2. Write down the value of \(u _ { 20 }\).

9. Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns: Row 1 □ Row 2 □ 1 Row 3 \includegraphics[max width=\textwidth, alt={}, center]{fff086fd-f5d8-45b7-8db1-8b22ba5aab31-11_40_104_566_479} She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make 2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.

1. Find an expression, in terms of \(n\), for the number of sticks required to make a similar arrangement of \(n\) squares in the \(n\)th row. Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.
2. Find the total number of sticks Ann uses in making these 10 rows. Ann started with 1750 sticks. Given that Ann continues the pattern to complete \(k\) rows but does not have sufficient sticks to complete the ( \(k + 1\) )th row,
3. show that \(k\) satisfies \(( 3 k - 100 ) ( k + 35 ) < 0\).
4. Find the value of \(k\).

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 \(a\).

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 .