What is a sequence:
- A set of numbers connected by a rule.
When working with any sequence, you must have:
- At least one term
- A rule
EG: Find the next three terms for the following arithmetic sequence: Un + 1 = Un + 4, U1 = 3
U2 = U1 + 4
U2 = 3 + 4 = 7 U2 = 7 |
U3 = U2 + 4
U3 = 7 + 4 = 11 U3 = 11 |
U4 = U3 + 4
U4 = 11 + 4 = 15 U4 = 15 |
Un Notation:
- Un = the nth term/general formula (U stands from the term and n stands for the number of the term)
- Un + 1 = the term after n/term to term rule/recurrence relationship (Un stands for the nth term and +1 stands for the following term)
- U1 = 1st term, U2 = 2nd term, U3 = 3rd term, U4 = 4th term ... Un = Nth term
Examples:
Find the next three terms and the rule for the following:
a) 14, 11, 8, 5, ...
Un + 1 = Un - 3 Therefore: ...5, 2, -1, -4 Un = -3n + 17 |
b) 1, 2, 4, 8, ...
Un + 1 = 2Un Therefore: ...8, 16, 32, 64 Un = 2^(n-1) |
c) 1, 3, 7, 15, ...
Un + 1 = 2Un + 1 Therefore: ...15, 31, 63, 127 Un = 2^n - 1 |
The Nth Terms:
- The Nth term is the general rule for a sequence.
- It allows us to find any term in the sequence by substituting the number of the term into where n is in the rule.
- Another way of writing the Nth term is by using the Un notation (above).
EG: U100 = -3(100) + 17 = -300 + 17 = -283
Term to Term Rule/Recurrence Relationships:
- Recurrence Relationships tell us the term to term rule for a sequence.
- This rule can only find the next term in a sequence.
- This can be done by substituting the previous term (which has to be known) into where Un is in the rule.
EG: U1 = 14, Un + 1 = Un - 3, U2 = U1 - 3 = 14 - 3 = 11
Examples:
Find the next three terms of the following sequences:
a) Un+1 = Un - 5, U1 = 43
U2 = U1 - 5 U2 = 43 - 5 = 38 U2 = 38 U3 = U2 - 5 U3 = 38 - 5 = 33 U3 = 33 U4 = U3 - 5 U4 = 33 - 5 = 28 U4 = 28 c) Un+1 = (Un + 2)/3, U1 = 31 U2 = (U1 + 2)/3 U2 = (31 + 2)/3 = 33/3 = 11 U2 = 11 U3 = (U2 + 2)/3 U3 = (11 + 2)/3 = 13/3 U3 = 13/3 U4 = (U3 + 2)/3 U4 = ((13/3) + 2)/3 = (13/3 + 6/3)/3 = (19/3)/3 = 19/9 U4 = 19/9 |
b) Un+1 = (Un - 3)/2, U1 = 47
U2 = (U1 - 3)/2 U2 = (47 - 3)/2 = 44/2 = 22 U2 = 22 U3 = (U2 - 3)/2 U3 = (22 - 3)/2 = 19/2 = 9.5 U3 = 19/2 U4 = (U2 - 3)/2 U4 = ((19/2) - 3)/2 = (9.5 - 3)/2 = 6.5/2 = 13/4 = 3.25 U4 = 13/4 d) Un+2 = Un+1 + Un, U1 = 1, U2 = 1 (FIBONACCI) U3 = U2 + U1 U3 = 1 + 1 = 2 U3 = 38 U4 = U3 + U2 U4 = 2 + 1 = 3 U4 = 3 U5 = U4 + U3 U5 = 3 + 2 = 5 U5 = 5 |
Types of Sequences:
- There are three types of sequences: Converging/Convergent, Diverging/Divergent, Oscilating/Periodic.
- Converging/Convergent: When n gets closer to infinity, Un gets closer to(or reaches a limit of) 0.
- Diverging/Divergent: When n gets closer to infinity, Un gets closer to infinity.
- Oscilating/Periodic: When n gets closer to infinity, Un fluctuates between a set of numbers, never getting closer to infinity and never reaching a limit of 0.
Examples of each type of Sequence:
Find the first four terms of the following sequences:
a) Un = 2^(-n)
U1 = 2^(-1) = 1/2 U2 = 2^(-2) = 1/(2²) = 1/4 U3 = 2^(-3) = 1/(2³) = 1/8 U4 = 2^(-4) = 1/(2^4) = 1/16 = 1/2, 1/4, 1/8, 1/16 As n gets closer to infinity, Un gets closer to 0, and so therefore the sequence is congergent. |
b) Un = 3n + 1
U1 = 3(1) + 1 = 3 + 1 = 4 U2 = 3(2) + 1 = 6 + 1 = 7 U3 = 3(3) + 1 = 9 + 1 = 10 U4 = 3(4) + 1 = 12 + 1 = 13 = 4, 7, 10, 13 As n gets closer to infinity, Un gets closer to infinity, and so therefore the sequence is divergent. |
c) Un = 5 + (-1)^n
U1 = 5 + (-1)¹ = 5 - 1 = 4 U2 = 5 + (-1)² = 5 + 1 = 6 U3 = 5 + (-1)³ = 5 - 1 = 4 U4 = 5 + (-1)^4 = 5 + 1 = 6 = 4, 6, 4, 6 As n gets closer to infinity, Un fluctuates between a set of numbers, and so therefore the sequence is oscilating or periodic. |
Finding the Limit (If There is a Limit):
- If a sequence is converging, then eventually, Un+1 = Un.
- So to find the limit, substitute Un+1 = x and Un = x
- Un+1 = (Un+4)/5 thus becomes x = (x+4)/5
- Now solve the equation: x = (x+4)/5 (x5), 5x = x + 4 (-x), 4x = 4 (/4), x = 1
- Test it by using U1 = 1: U2 = (U1+4)/5 = (1+4)/5 = 5/5 = 1
- The sequence will converge on 1, so it's limit is 1.