1. What is a sequence?
MX233 and College algebra

2. A sequence is a set of numbers arranged in a definite pattern
1, 3, 5, 7, 9, … a rule is ‘add 2’
1, 2, 4, 8, …a rule is ‘multiply by 2’
2, 4, 6, 8, …a rule is ‘add 2 to the numerator and denominator

3. Write down a rule and the next three terms for the following
3, 7, 11, 15,…
a, 3a, 5a, 7a,…
16, 8, 4, 2,…
3, 9, 27, 81,…
3 , 4 , 5 , 6 ,…
Add 4; 19, 23, 27
Add 2a; 9a, 11a, 13a
Halve or multiply by ½ ; 1, ½, ¼
Multiply by 3, 243, 729, 2187
Add 1 to numerator and add 2 to denominator
7 , 8 , 9

4. The symbol for the nth term is tn or Tn
Each term of a sequence is associated with the corresponding element of the set of natural numbers
N = {1, 2, 3, 4, …}
The symbol for the nth term is tn or Tn

5. The general term of a sequence is referred to by a letter of the alphabet:
an ‘the nth term’
tr ‘the rth term’
ui ‘the ith term’
all name the general term

6. General Sequences
Give the first three terms of the following sequences with the general term Tn
Tn = 1 n2 10
2. Tn = 2n-1
0.1, 0.4, 0.9
1, 2, 4

7. Arithmetic progression or sequence
A sequence following the rule ‘add a fixed amount’ is called an arithmetic sequence or arithmetic progression, AP for short
For the sequence 2, 5, 8, 11, 14,…
The first term t1 is 2
The second term is t2 is 5
t3 is 8
What is t8 or t25?

8. nth or General Term of an Arithmetic Sequence
where a1 is the first term of the sequence and d is the common difference.
Where d = a2-a1 = a3-a2

9. Find the first five terms and the15th term of the arithmetic sequence
Note how the first term is 3. What was the common difference for this arithmetic sequence?
If you said ½, you are correct!  Note how each term went up by 1/2 from the previous term.

10. Find the first five terms and the15th term of the arithmetic sequence
Note how the first term is not -5, but -21/4.  The formula for the nth term used n, instead of n - 1 that is in the general form
If you said -1/4 you are correct!  Note how each term went DOWN by 1/4 from the previous term.

11. Write a formula for the nth term of the arithmetic sequence -10, -5, 0, 5, ....
d, the common difference?
= -5, = 0, and = 5.   It has to be consistent throughout the sequence.

12. Write a formula for the nth term of the arithmetic sequence
What is a1, the first term of the sequence?
The first term of this sequence is
What is d, the common difference?
Note that you would have to add   to each previous term to get to the next term.

13. Find the twentieth term of the AP: 3, 8, 13, 18, …
Here, a = 3 and d = 5
We want t20 = 3 + (20 – 1)5
= 98

14. Find the fifteenth term of the AP: 20, 15, 10, 5, …
Here, a = 20 and d = -5 (t2-t1 = 15-20)
We want t15 = 20 + (15 – 1)(-5)
= -50