Worked examples
For each of the following patterns, find the general rule. Write this rule in algebraic language. 1 1; 3; 5; 7; …
2
Solutions 1 Tn = 2n − 1
__ 3 ; 1
1
__ 4 ; 1
__ 5 ; 1
__ 6 ; …
3 0; 3; 8; 15; …
Write the pattern in terms of the number of each term, as follows: 1st term: 1 = 2(1) – 1 2nd term: 3 = 2(2) – 1 3rd term: 5 = 2(3) – 1 and so on. So, the nth term = 2(n) – 1 = 2n – 1
2 Tn = 1
____ 2 + n or 1
____ n + 2
Write the pattern in terms of the number of each term, as follows: 1st term: 1 2nd term: 1 3rd term: 1
__ 3 = 1
____ 2 + 1
__ 4 = 1
__ 5 = 1
3 Tn = n2 − 1
____ 2 + 2
____ 2 + 3 and so on.
So, the nth term = 1
____ 2 + n
Write the pattern in terms of the number of each term, as follows: 1st term: 0 = 12 – 1 2nd term: 3 = 22 – 1 3rd term: 8 = 32 – 1 and so on. So, the nth term = n2 – 1
Once we have the rule for the general term, or nth term, we can find any term in the pattern − we do not have to calculate all the terms in between until we get to the term that we want.
The common difference When there is a common difference between successive terms in a number
pattern, the general term will be of the form Tn = an + c, where a is the common difference and c is a constant.
In the first example above, you saw that there is a common difference of 2 between successive terms in the pattern.
The formula for Tn in this case is Tn = 2n − 1, where the coefficient of n is 2, the common difference, and −1 is the constant. This is a very useful tip to remember.
General term equation
Tn = an + c where a is the common difference and c is a constant.
Unit 2: The general rule
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