Series Sequences Piecewise Functions Sequences and Series Alg

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 = a 1 + (n-1) * d • • an : term you are looking for n : number of the term in the sequence a 1 : 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 8 th term of 3, 7, 11, …

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

Arithmetic Sequences • Given two terms – find the Explicit Formula – Find the common difference – Use one given term & common difference to find a 1 – Input a 1 & d into formula – leaving an and n • Example: a 9 = 120; a 14 = 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: S 8 for 3, 6, 9, … – First find 8 th term then find S 8

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 = a 1* rn-1 • • an : term you are looking for n : number of the term in the sequence a 1 : 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 8 th term – an = -2 * 2 n-1 • Given two terms – find the Explicit Formula – Find the common ratio – Use one given term & common ratio to find a 1 – Input a 1 & r into formula – leaving an and n • Example: a 3 = 36; a 5 = 324 • .

Arithmetic & Geometric Mean • Arithmetic mean – The middle of an arithmetic sequence – the average – (a 1 + 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: S 7 for 3 – 6 + 12 – 24. . . • Example:

Infinite Geometric Series • Series convergence – Find common ratio (r) – If Ir. I > 1, series diverges (goes to infinity) – If Ir. I < 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 = a 1 + (n-1) * d • • an is the term looked for a 1 is the first term n is the number of the term d is the common difference between terms • Geometric – Nth term : an = a 1*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 = a 1 + (n-1) * d • Geometric : an = a 1*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