Series & Sequences Piecewise Functions

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

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, …

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

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

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

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:

Geometric Sequences (Alg 2 – 9.4-5)

• Exponential – common ratio between terms– Example: 1, -5, 25, -125, … 1/2, 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)

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

• .

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

.

Partial Sum (Geometric)

• Partial Sum (n terms of sequence):

• Example: S7 for 3 – 6 + 12 – 24 ...

• Example:

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:

: 5+4+3.2+2.56

Formulas

• Recursive: Using a Previous Term• Arithmetic– 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

Formulas

• Explicit: Finding the nth Term• Arithmetic– 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

Equations (Both Explicit & Recursive)

• 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

Sum (A) or Partial Sum (G)

• Summation: Symbol is the Greek letter Σ• Partial Sum: Symbol is Sn

• For Arithmetic :

• For Geometric :

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