QUESTION 4
19]
Given the linear pattern: 2x + 1; 3x + 3; 4x + 5;...
Write down the next term in the pattern.
(1)
Write, in terms of x, the formula for T, the general term of
(3)
the pattern.
If the value of the 13th term of the pattern is 95, calculate the
(2)
value of x.
4.
If x=5, determine the largest value of n for which Tn < 158.
(3)
12 ) Page
Answers
Answer:
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Answer:
It is possible to determine a formula for linear sequences, i.e. sequences where the difference between successive terms is always the same.
The first differences for the number pattern
11 14 17 20 23 26 ...
are 3 3 3 3 3
If we look at the sequence 3n, i.e. the multiples of 3, and compare it with our original sequence
our sequence 11 14 17 20 23 26
sequence 3n 3 6 9 12 15 18
we can see easily that the formula that generates our number pattern is
nth term of sequence = 3n + 8
i.e. un = 3n + 8
If, however, we had started with the sequence
38 41 44 47 50 53 ...
the first differences would still have been 3 and the comparison of this sequence with the sequence 3n
our sequence 38 41 44 47 50 53
sequence 3n 3 6 9 12 15 18
would have led to the formula un = 3n + 35.
In the same way, the sequence
–7 –4 –1 2 5 8 ...
also has first differences 3 and the comparison
our sequence – 7 – 4 – 1 2 5 8
sequence 3n 3 6 9 12 15 18
yields the formula un = 3n – 10.
From these examples, we can see that any sequence with constant first difference 3 has the formula
un = 3n + c
where the adjustment constant c may be either positive or negative.
This approach can be applied to any linear sequence, giving us the general rule that:
If the first difference between successive terms is d, then
un = d × n + c
Example 1
Determine a formula for this sequence:
7, 13, 19, 25, 31, ...
Example 2
Determine a formula for this sequence:
2, 7, 12, 17, 22, 27, ...
Example 3
Determine a formula for the sequence:
28, 25, 22, 19, 16, 13, ...
Exercises
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Question 1
For the sequence,
7, 11, 15, 19, ...
(a)
calculate the difference between successive terms,
(b)
determine the formula that generates the sequence.
un =
Question 2
Determine the formula for each of the following sequences:
(a) 6, 10, 14, 20, 24, ... un =
(b) 11, 13, 15, 17, 19, ... un =
(c) 9, 16, 23, 30, 37, ... un =
(d) 34, 56, 78, 100, 122, ... un =
(e) 22, 31, 40, 49, 58, ... un =
Question 3
One number is missing from the following sequence:
1, 6, 11,
, 21, 26, 31
(a)
Write in the missing number.
(b)
Calculate the difference between successive terms.
(c)
Determine the formula that generates the sequence.
un =
Question 4
Determine the general formula for each of the following sequences:
(a) 1, 4, 7, 10, 13, ... un =
(b) 2, 6, 10, 14, 18, ... un =
(c) 4, 13, 22, 31, 40, ... un =
(d) 5, 15, 25, 35, 45, ... un =
(e) 1, 20, 39, 58, 77, ... un =
Question 5
For the sequence,
18, 16, 14, 12, 10, ...
(a)
calculate the difference between successive terms,
(b)
determine the formula that generates the sequence.
un =
Question 6
Determine the general formula for each of the following sequences:
(a) 19, 16, 13, 10, 7, ... un =
(b) 100, 96, 92, 88, 84, ... un =
(c) 41, 34, 27, 20, 13, ... un =
(d) 66, 50, 34, 18, 2, ... un =
(e) 90, 81, 72, 63, 54, ... un =
Question 7
For the sequence,
–2, –4, –6, –8, –10, –12, ...
(a)
calculate the difference between successive terms,
(b)
determine the formula for the sequence.
un =
Question 8
Determine the formula that generates each of the following sequences:
(a) 0, – 5, – 10, – 15, – 20, ... un =
(b) – 18, – 16, – 14, – 12, – 10, ... un =
(c) – 5, – 8, – 11, – 14, – 17, ... un =
(d) 8, 1, – 6, – 13, – 20, ... un =
(e) – 7, – 3, 1, 5, 9, ... un =
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