1.04e Sequences: nth term and recurrence relations

12 Calculate \(\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }\). 12 A sequence begins $$\begin{array} { l l l l l l l l l l l } 1 & 3 & 5 & 3 & 1 & 3 & 5 & 3 & 1 & 3 & \ldots \end{array}$$ and continues in this pattern.

1. Find the 55th term of this sequence, showing your method.
2. Find the sum of the first 55 terms of the sequence.

2 The terms of a sequence are given by $$\begin{aligned} u _ { 1 } & = 192 , \\ u _ { n + 1 } & = - \frac { 1 } { 2 } u _ { n } . \end{aligned}$$

1. Find the third term of this sequence and state what type of sequence it is.
2. Show that the series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) converges and find its sum to infinity.

3 A sequence begins $$\begin{array} { l l l l l l l l l l l l } 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 & 1 & \ldots \end{array}$$ and continues in this pattern.

1. Find the 48th term of this sequence.
2. Find the sum of the first 48 terms of this sequence.

4 Sequences A, B and C are shown below. They each continue in the pattern established by the given terms.

1. Which of these sequences is periodic?
2. Which of these sequences is convergent?
3. Find, in terms of \(n\), the \(n\)th term of sequence A .

4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 3 ^ { 2 n } - 1$$

1. Write down the value of \(u _ { 1 }\).
2. Show that \(u _ { n + 1 } - u _ { n } = 8 \times 3 ^ { 2 n }\).
3. Hence prove by induction that each term of the sequence is a multiple of 8 .

6 A sequence is defined by \(a _ { 1 } = 7\) and \(a _ { k + 1 } = 7 a _ { k } - 3\).

1. Calculate the value of the third term, \(a _ { 3 }\).
2. Prove by induction that \(a _ { n } = \frac { \left( 13 \times 7 ^ { n - 1 } \right) + 1 } { 2 }\).

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.

4.A sequence of positive integers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has \(r\) th term given by $$a _ { r } = 2 ^ { r } - 1$$

1. Write down the first 6 terms of this sequence.
2. Verify that \(a _ { r + 1 } = 2 a _ { r } + 1\)
3. Find \(\sum _ { r = 1 } ^ { n } a _ { r }\)
4. Show that \(\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }\)
5. Hence show that \(1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)\)
6. Show that \(\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }\)

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

3 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 24 - \frac { 2 } { 3 } n$$

1. Write down the exact values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find the value of \(k\) such that \(u _ { k } = 0\).
3. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).

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 \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by \(u _ { n } = 3 n + 2\) for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. State what type of sequence \(S\) is.
3. Find \(\sum _ { n = 101 } ^ { 200 } u _ { n }\).

6 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 85 - 5 n\) for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
3. Given that \(u _ { 1 } , u _ { 5 }\) and \(u _ { p }\) are, respectively, the first, second and third terms of a geometric progression, find the value of \(p\).
4. Find the sum to infinity of the geometric progression in part (iii).

4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 5 n + 1\).

1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Evaluate \(\sum _ { n = 1 } ^ { 40 } u _ { n }\). Another sequence \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { 1 } = 2\) and \(w _ { n + 1 } = 5 w _ { n } + 1\).
3. Find the value of \(p\) such that \(u _ { p } = w _ { 3 }\).

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 \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 3 n - 1\), for \(n \geqslant 1\).

1. Find the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find \(\sum _ { n = 1 } ^ { 40 } u _ { n }\).

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

10 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 3\) and \(u _ { n + 1 } = 3 u _ { n } - 2\).

1. Find \(u _ { 2 }\) and \(u _ { 3 }\) and verify that \(\frac { 1 } { 2 } \left( u _ { 4 } - 1 \right) = 27\).
2. Hence suggest an expression for \(u _ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.

6 A sequence is defined by \(a _ { 1 } = 1\) and \(a _ { k + 1 } = 3 \left( a _ { k } + 1 \right)\).

1. Calculate the value of the third term, \(a _ { 3 }\).
2. Prove by induction that \(a _ { n } = \frac { 5 \times 3 ^ { n - 1 } - 3 } { 2 }\).

6 A sequence is defined by \(u _ { 1 } = 3\) and \(u _ { n + 1 } = 3 u _ { n } - 5\). Prove by induction that \(u _ { n } = \frac { 3 ^ { n - 1 } + 5 } { 2 }\). Section B (36 marks)

6 A sequence is defined by \(u _ { 1 } = 8\) and \(u _ { n + 1 } = 3 u _ { n } + 2 n + 5\). Prove by induction that \(u _ { n } = 4 \left( 3 ^ { n } \right) - n - 3\).

9 For the sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), it is given that \(u _ { 1 } = 8\) and $$u _ { r + 1 } = \frac { 5 u _ { r } - 3 } { 4 }$$ for all \(r\).

1. Prove by mathematical induction that $$u _ { n } = 4 \left( \frac { 5 } { 4 } \right) ^ { n } + 3$$ for all positive integers \(n\).
2. Deduce the set of values of \(x\) for which the infinite series $$\left( u _ { 1 } - 3 \right) x + \left( u _ { 2 } - 3 \right) x ^ { 2 } + \ldots + \left( u _ { r } - 3 \right) x ^ { r } + \ldots$$ is convergent.
3. Use the result given in part (i) to find surds \(a\) and \(b\) such that $$\sum _ { n = 1 } ^ { N } \ln \left( u _ { n } - 3 \right) = N ^ { 2 } \ln a + N \ln b .$$

5 The cubic equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers, has roots \(\alpha , \beta\) and \(\gamma\), such that $$\begin{aligned} \alpha + \beta + \gamma & = 15 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 \end{aligned}$$ Write down the value of \(p\) and find the value of \(q\). Given that \(\alpha , \beta\) and \(\gamma\) are all real and that \(\alpha \beta + \alpha \gamma = 36\), find \(\alpha\) and hence find the value of \(r\).