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| nth Terms and Sequences |
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Sequences
A number sequence is a set of numbers (terms) in which a pattern can be seen and a rule can be used to find every term in the sequence.
For example: |
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5, 10, 20, 40… double the last term each time…..80, 160
3, 5, 7, 9,…..Add two each time…..11, 13
25, 21, 17, 13,…minus four each time…..9, 5
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| The above examples have simple patterns, with harder sequences we need to look for a pattern and then establish the rule in order to calculate any term in the sequence. |
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nth Term
The rule for finding any term is called the nth term.
For example:
Given the sequence 6, 10, 14, 18,…… |
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| a) Find the nth term; b) Find the 20th term; c) If the nth term is 42, what is the value of n? |
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We look at the differences between each term:
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6 10 14 18
\ _/\_ /\_ /
4 4 4 |
The difference is four. |
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The general formula for the nth term is:
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where a = the first term = 6,
n = the number of the term, and
d = the difference = 4. |
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| a) For this sequence: |
nth term = 6 + (n – 1) 4
= 6 + 4n – 4
= 2 + 4n |
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| We can now use this formula to work out the value of any term in the sequence. |
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b) |
20th term =
2 + 4 x 20
= 82 |
because n = 20 |
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c) |
nth term = 42 |
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42 = 2 + 4n
40 = 4n
n = 10 |
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So the 10th term is 42.
This formula will work for any linear sequence. In a linear sequence, the difference is constant. 4 in the sequence above. |
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Quadratic Sequences |
In this type, the first difference is not constant. The second difference gives a constant.
For example: 3, 8, 15, 24, 36……… is a sequence. |
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3 8 15 24 35
\_/\_ /\_ /\_ /
5 7 9 11
\_ /\_ /\_ /
2 2 2 |
(first difference)
(second difference) |
This is a quadratic sequence as the second difference is a constant (in this case, 2).
The general formula for a quadratic is: |
| nth term = a + (n – 1)d1 + ½(n – 1)(n – 2)d2 |
| Where a = first term |
d1= first difference |
d2= second difference |
| = 3 |
= 5 |
= 2 |
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nth term = 3 + (n – 1)5 + ½ (n – 1)(n – 2)2
= 3 + 5n – 5 + n2 – 3n + 2
= 2n + n2 |
We can use this to find the 100th term:
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100th term = 1 002 + 200
= 10 200 |
Provided that the second difference is a constant we can use this method for any quadratic sequence.
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