Sequences and Series

By: Brandon Huggins

Brad Norvell

Andrew Knight

Arithmetic and Geometric Series

• Arithmetic– 1, 5, 9, 13, 17, 21

• +4 +4 +4 +4 +4

– 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 = a1 / (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