1. Series & Sequences Piecewise Functions

2. Sequences and Series (Alg 2 – 9.1)
Sequence – ordered set of numbers
Arithmetic - Linear
Common difference between consecutive terms
Geometric - Exponential
Common ratio between consecutive terms
Terminology
n : the number of the term in the sequence
a : the actual number or constant
Sequence : an ordered set of numbers (n, an)
Series : sum of the number of terms

3. Arithmetic Sequences (Alg 2 – 9.3)
Linear – common difference between terms
General rules (formulas) for sequence
Explicit: Any Term: an = a1 + (n-1) * d
an : term you are looking for
n : number of the term in the sequence
a1 : first term of the sequence
d : common difference
Recursive: Next Term: an = an-1 + d
an & d same as above
an-1 is the previous term (the one just before an)
Finding the nth term example:
Find 8th term of 3, 7, 11, …

4. Sequences & Series Finding terms based on Explicit Formula
Example: an = 3n – 5
Find 1st 5 terms (n = 1, 2, 3, 4, 5)
Find the 10th term

5. Arithmetic Sequences Given two terms – find the Explicit Formula
Find the common difference
Use one given term & common difference to find a1
Input a1 & d into formula – leaving an and n
Example: a9 = 120; a14 = 195
If a term is missing in the middle
First find average difference
Then add/subtract difference from known
Continue until gap filled
Example: 2, __, __, __, 18

6. Partial Sum
Indicates how many terms, from the beginning of a sequence, to add for the result
Indicated by Sn.
Arithmetic Partial Sum:
Example: S8 for 3,6,9,…
First find 8th term then find S8

7. Summation Notation Indicated by Greek letter Σ
The sum of the indicated number of terms using the identified formula
Bottom number is start place
Top number is end place
ak stands for the formula used
Example:

8. Geometric Sequences (Alg 2 – 9.4-5)
Exponential – common ratio between terms
Example: 1, -5, 25, -125, … /2, 1/4, 1/8, 1/16, …
General rules (formulas) for sequence
Explicit: Any Term: an = a1* rn-1
an : term you are looking for
n : number of the term in the sequence
a1 : first term of the sequence
r : common ratio
Recursive: Next Term: an = an-1 * r
an & r same as above
an-1 is the previous term (the one just before an)

9. Geometric Sequences Finding the nth term example:
Given the explicit formula, find first 4 terms and 8th term
an = -2 * 2n-1
Given two terms – find the Explicit Formula
Find the common ratio
Use one given term & common ratio to find a1
Input a1 & r into formula – leaving an and n
Example: a3 = 36; a5 = 324 .

10. Arithmetic & Geometric Mean
Arithmetic mean
The middle of an arithmetic sequence – the average
(a1 + an) ÷ 2
Example: 3, 6, 9, 12, 15, 18, 21, 24, 27
Geometric mean
The value of a term between 2 non-consecutive terms in a geometric sequence
Found by multiplying the terms and then taking the square root : geometric mean = √(a*b)
Example: Find geometric mean of 16 and 25 .

11. Partial Sum (Geometric)
Partial Sum (n terms of sequence):
Example: S7 for 3 – –
Example:

12. Infinite Geometric Series
Series convergence
Find common ratio (r)
If IrI > 1, series diverges (goes to infinity)
If IrI < 1, series converges (moves towards a limit)
If a series converges – you can find the limit
Example: :

13. Formulas Recursive: Using a Previous Term Arithmetic Geometric
Next term : an = an-1 + d
an is the term looked for
an-1 is the previous term
d is the common difference between terms
Geometric
Next term : an = an-1*r
Same as above except r is the common ratio

14. Formulas Explicit: Finding the nth Term Arithmetic Geometric
Nth term : an = a1 + (n-1) * d
an is the term looked for
a1 is the first term
n is the number of the term
d is the common difference between terms
Geometric
Nth term : an = a1*rn-1
Same as above except r is the common ratio

15. Equations (Both Explicit & Recursive)
Finding the next term based on a previous term
Arithmetic : an = an-1 + d
Geometric : an = an-1 * r
Explicit
Finding a term based on the first term and the change
Arithmetic : an = a1 + (n-1) * d
Geometric : an = a1*rn-1

16. Sum (A) or Partial Sum (G)
Summation: Symbol is the Greek letter Σ
Partial Sum: Symbol is Sn
For Arithmetic :
For Geometric :

17. Piecewise Functions
Function has different equation depending on the input value
Example: Constant Value
Tickets: <10, $5 each; 10 – 20, $4 each; >20, $3 each
Can work with any type of pieces of functions
Basis for most real-world modeling