Sequences and Series By: Brandon Huggins Brad Norvell Andrew Knight

Arithmetic and Geometric Series Arithmetic – 1, 5, 9, 13, 17, –Each number is added or subtracted Geometric – 1, 2, 4, 8, 16, 32 x2 x2 x2 x2 x2 –Each number is multiplied or divided

Recursive Formulas The easiest way to define a series What you do to the current term to get to the next term Arithmetic: 1,3,5,7,9... –a n+1 = a n + 2 Geometric: 1,2,4,8,16... –a n+1 = 2a n

Finding a Term in an Arithmetic Sequence Formula= a subscript 1 is the first term of the sequence d is the common difference n is the number of the term to find

Finding a term in a geometric sequence Formula= a subscript 1 is the first term of the sequence r is the common ratio n is the number of the term to find Limit= 0 And Infinity+1

Limits of sequences Arithmetic sequences cannot have a limit Geometric can, but only if the common ratio is between -1 and 1 Limit is 0 If arithmetic, or if common ratio is less than -1 or greater than 1, the limit is infinity

Sum of an Arithmetic Series This is the formula to add all of the numbers of the series before the designated number= Sn is the sum of n terms or nth partial sum a subscript 1 is the first term a subscript n is the term that you want to go to n is the number of the term you want to find

Sum of an Geometric Series This is the formula to add all of the numbers of the series before the designated number= Sn is the sum of n terms or nth partial sum a subscript 1 is the first term r is the common ratio n is the number of the term you want to find

Mathematical Induction Proving summation formula Just watch the example

Sigma Notation Formula= n is the number that you increase the number in parenthesis by The number atop the E looking writing is the number you go to The E symbol means to add all of the solutions together

Infinite Sums For arithmetic, it is always infinity For geometric, the common ratio must be between -1 and 1. The formula is S = a 1 / (1 – x) Similar to the geometric sum formula

Compound Interest Formula=A = P (1 + r/n) to the (nt) P = the original investment r = annual interest rate as a percentage n = the number of times per year interest is compounded t = the length of the term (investment or loan) A = the amount accumulated after n periods

Application of Summations Can be used in everyday life Population is a common application Most are just simple arithmetic or geometric sequences. Infinite sums are not as commonly used