Post on 01-Apr-2015
transcript
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