Sequences
Sequence – a list of numbers in a given order
Terms – the numbers in a sequence
Consecutive terms – terms next to one another in a sequence
A sequence is a pattern of numbers, connected by a rule.
Arithmetic sequences
In arithmetic sequences, the difference between consecutive terms is a constant or fixed number. Arithmetic sequences are sometimes referred to as linear sequences.
For example, the sequence 3, 5, 7, 9, … is an arithmetic sequence with a constant difference of 2 between consecutive terms.
Finding the nth term of an arithmetic sequence
Find the difference between consecutive terms.
If the difference is 2 then your expression for the nth term will include 2n, if the difference is 3 the expression for the nth term will include 3n
Compare the sequence with the values of 2n or difference x n to find what you need to add or subtract to give your final expression for the nth term.
Using the example above:
Sequence 3 5 7 9 11
2n 2 4 6 8 10
Each term in the sequence is 1 more than 2n, so the expression for the nth term is 2n + 1
What if the numbers in the sequence are going down instead of up?
The same principles apply.
For example, take the sequence 50, 46, 42, 38, 34, …
The difference between consecutive terms is –4, so the expression for the nth term contains –4n
Sequence 50 46 42 38 34
-4n -4 -8 -12 -16 -20
Each term in the sequence is 54 more than –4n, so the expression for the nth term is –4n + 54 or 54 – 4n.
Quadratic sequences
In a quadratic sequence, the difference between consecutive terms is not constant.
For example, the sequence 2, 5, 10, 17, 26, … is a quadratic sequence. The first differences are not constant, 3, 5, 7, 9, … so, now it’s time to look at the second differences. Second differences are the differences between consecutive (first) differences.
Sequence 2 5 10 17 26
1st Differences 3 5 7 9
2nd Differences 2 2 2
The second differences are constant, 2. The expression for the nth term contains n2.
Compare the sequence to n2
Sequence 2 5 10 17 26
n2 1 4 9 16 25
Each term in the sequence is 1 more than n2, so the expression for the nth term is n2 + 1
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