Recurrence relation evaluation

7 A sequence is defined by the recurrence relation \(\mathrm { u } _ { \mathrm { k } + 1 } = \mathrm { u } _ { \mathrm { k } } + 5\) with \(\mathrm { u } _ { 1 } = - 2\).

1. Write down the values of \(u _ { 2 } , u _ { 3 }\), and \(u _ { 4 }\).
2. Explain whether this sequence is divergent or convergent.
3. Determine the value of \(u _ { 30 }\).
4. Determine the value of \(\sum _ { \mathrm { k } = 1 } ^ { 30 } \mathrm { u } _ { \mathrm { k } }\).

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 \(u _ { 5 }\).

7. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation $$u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000 .$$ In this relation \(u _ { n }\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year. Given that \(d = 15000\),

1. calculate \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\) and comment briefly on your results. Given that \(d = 100000\),
2. show that the population of fish dies out during the sixth year.
3. Find the value of \(d\) which would leave the population each year unchanged.

6. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation \(\quad u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000\).
In this relation \(u _ { n }\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year.
Given that \(d = 15000\),

1. calculate \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\) and comment briefly on your results. Given that \(d = 100000\),
2. show that the population of fish dies out during the sixth year.
3. Find the value of \(d\) which would leave the population each year unchanged.

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

$$u _ { n + 1 } = \left( u _ { n } \right) ^ { 2 } - 1 , \quad n \geq 1 .$$ 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. 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.
2. Evaluate $$\sum _ { r = 1 } ^ { 20 } u _ { r } .$$

4. 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 \(b\).

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 \(k\).

6. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots , u _ { n }\) is defined by the recurrence relation $$u _ { n + 1 } = p u _ { n } + 5 , u _ { 1 } = 2 , \text { where } p \text { is a constant. }$$ Given that \(u _ { 3 } = 8\),

1. show that one possible value of \(p\) is \(\frac { 1 } { 2 }\) and find the other value of \(p\). Using \(p = \frac { 1 } { 2 }\),
2. write down the value of \(\log _ { 2 } p\). Given also that \(\log _ { 2 } q = t\),
3. express \(\log _ { 2 } \left( \frac { p ^ { 3 } } { \sqrt { q } } \right)\) in terms of \(t\).
   [0pt] [P2 November 2002 Question 4]

1. 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 \(u _ { 20 }\).

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 \(u _ { 20 }\).

12. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation $$u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000 .$$ In this relation \(u _ { n }\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year. Given that \(d = 15000\),

1. calculate \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\) and comment briefly on your results. Given that \(d = 100000\),
2. show that the population of fish dies out during the sixth year.
3. Find the value of \(d\) which would leave the population each year unchanged.