Series & Sequences

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 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

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

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

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