1. Higher Maths Sequences Strategies Click to start
Higher Maths
Strategies
Sequences
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2. Sequences The following questions are on
Maths4Scotland Higher
The following questions are on
Sequences
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3. Limit = 16½ Maths4Scotland Higher
A recurrence relation is defined by where -1 < p < -1 and u0 = 12
a) If u1 = and u2 = find the values of p and q
b) Find the limit of this recurrence relation as n  
Put u1 into recurrence relation
Put u2 into recurrence relation
Solve simultaneously:
(2) – (1)
Hence
State limit condition
-1 < p < 1, so a limit L exists
Hint
Use formula
Limit = 16½
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4. un = height at the start of year
Maths4Scotland Higher
A man decides to plant a number of fast-growing trees as a boundary between his property and the property of
his neighbour. He has been warned however by the local garden centre, that during any year, the trees
are expected to increase in height by 0.5 metres.
In response to this warning, he decides to trim 20% off the height of the trees at the start of any year.
If he adopts the “20% pruning policy”, to what height will he expect the trees to grow in the long run.
His neighbour is concerned that the trees are growing at an alarming rate and wants assurance
that the trees will grow no taller than 2 metres. What is the minimum percentage that the trees
will need to be trimmed each year so as to meet this condition.
un = height at the start of year
Construct a recurrence relation
State limit condition
-1 < 0.8 < 1, so a limit L exists
Use formula
Limit = 2.5 metres
Use formula again
Hint
m = 0.75
Minimum prune = 25%
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5. u0 = 2500 Maths4Scotland Higher Construct a recurrence relation
On the first day of March, a bank loans a man £2500 at a fixed rate of interest of 1.5% per month.
This interest is added on the last day of each month and is calculated on the amount due on the first day of the
month. He agrees to make repayments on the first day of each subsequent month. Each repayment is £300
except for the smaller final amount which will pay off the loan.
 a) The amount that he owes at the start of each month is taken to be the amount still owing just after the
monthly repayment has been made.
Let un and un+1 and represent the amounts that he owes at the starts of two successive months.
Write down a recurrence relation involving un and un+1
b) Find the date and amount of the final payment.
u0 = 2500
Construct a recurrence relation
Calculate each term in the recurrence relation
1 Mar u0 =
1 Apr u1 =
1 May u2 =
1 Jun u3 =
1 Jul u4 =
1 Aug u5 =
1 Sept u6 =
1 Oct u7 =
1 Nov u8 =
1 Dec Final payment £290.68
Hint
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6. Limit = 25 Maths4Scotland Higher
Two sequences are generated by the recurrence relations and
The two sequences approach the same limit as n  .
Determine the value of a and evaluate the limit.
Use formula for each sequence
Sequence 1
Sequence 2
Equate the two limits
Cross multiply
Simplify
Solve
Hint
Deduction
Since limit exists a  1, so
Limit = 25
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7. Two sequences are defined by the recurrence relations
Maths4Scotland Higher
Two sequences are defined by the recurrence relations
If both sequences have the same limit, express p in terms of q.
Use formula for each sequence
Sequence 1
Sequence 2
Equate the two limits
Cross multiply
Rearrange
Hint
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8. Two sequences are defined by these recurrence relations
Maths4Scotland Higher
Two sequences are defined by these recurrence relations
a) Explain why only one of these sequences approaches a limit as n  
b) Find algebraically the exact value of the limit.
c) For the other sequence find
i) the smallest value of n for which the nth term exceeds 1000, and
ii) the value of that term.
First sequence has no limit since 3 is not between –1 and 1
Requirement for a limit
2nd sequence has a limit since –1 < 0.3 < 1
Sequence 2
u0 = 1
u1 =
u2 =
u3 =
u4 = 65
u5 =
u6 =
u7 =
List terms of 1st sequence
Hint
Smallest value of n is 8; value of 8th term =
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9. You have completed all 6 questions in this presentation
Maths4Scotland Higher
You have completed all 6 questions in this presentation
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10. Maths4Scotland Higher
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