1. SEQUENCES
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2. A sequence is a set of terms, in a definite order, where the terms are obtained by some rule.
A finite sequence ends after a certain number of terms.
An infinite sequence is one that continues indefinitely.

3. For example: 1, 3, 5, 7, … 1st term = 2 x 1 – 1 = 1
(This is a sequence of odd numbers)
1st term = 2 x 1 – 1 = 1
+ 2
2nd term = 2 x 2 – 1 = 3
+ 2
3rd term = 2 x 3 – 1 = 5
. .
nth term = 2 x n – 1 = 2n - 1

4. NOTATION 1st term = u 2nd term = u 3rd term = u . . nth term = u 1 2 3
. .
nth term = u n

5. OR
1st term = u
2nd term = u 1
3rd term = u 2
. .
nth term = u
n-1

6. FINDING THE FORMULA FOR THE TERMS OF A SEQUENCE

7. A recurrence relation defines the first term(s) in the sequence and the relation between successive terms.

8. For example: 5, 8, 11, 14, … u = 5 u = u +3 = 8 u = u +3 = 11 .2 1
u = u +3 = 11 3 2 .
u = u +3 = 3n + 2
n+1 n

9. when looking for the rule
What to look for
when looking for the rule
defining a sequence

10. Constant difference: coefficient of n is the difference
2nd level difference: compare with square numbers
(n = 1, 4, 9, 16, …)
3rd level difference: compare with cube numbers
(n = 1, 8, 27, 64, …)
None of these helpful: look for powers of numbers
(2 = 1, 2, 4, 8, …)
Signs alternate: use (-1) and (-1)
-1 when k is odd +1 when k is even 2 3
n - 1 k k

11. EXAMPLE:
Find the next three terms in the sequence 5, 8, 11, 14, …

12. EXAMPLE: The nth term of a sequence is given by x =
Find the first four terms of the sequence.
b) Which term in the sequence is ?
c) Express the sequence as a recurrence relation. 1 __ n n 2 1
____
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13. EXAMPLE:
Find the nth term of the sequence +1, -4, +9, -16, +25, …

14. EXAMPLE: A sequence is defined by a recurrence relation of the form:
M = aM + b.
Given that M = 10, M = 20, M = 24, find the value of a and the
value of b and hence find M .
n + 1 1 2 3 4