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Sequences and Series By: Brandon Huggins Brad Norvell Andrew Knight

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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

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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

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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

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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

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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

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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

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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

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Mathematical Induction Proving summation formula Just watch the example

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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

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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

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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

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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

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