Sequences and Series ARITHMETIC SEQUENCES Alana Poz Sequence

Sequence � Sequence – ordered progression of numbers �There is always a rule or

Difference Between Arithmetic and Geometric Sequences � Arithmetic Sequence – the terms have a

Explaining “n” n = index number � This means that whatever n equals, is

Recursive Formulas*used to find next term �Arithmetic Recursive Formula �an = an– 1 +

Practice Recursive Formulas ex) Given: 10, 12, 14… Find the next term. 1. Use

Explicit Formulas *used to calculate any number in the sequence � Arithmetic Explicit Formula

Practice Explicit Formulas � Arithmetic Practice ex) Given: -7, -1, 5, 11… Find n

Series = Sum of a Sequence � Arithmetic Summation Formula �This formula calculates the

Practice Summation Formulas � Arithmetic Practice ex) Given: 117, 110, 103… 33 sum. Find

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Sequences and Series ARITHMETIC SEQUENCES Alana Poz

Sequence � Sequence – ordered progression of numbers �There is always a rule or the ability to generate a rule to describe a sequence

Difference Between Arithmetic and Geometric Sequences � Arithmetic Sequence – the terms have a common difference �The difference between each term will always be the same and is the amount between each term ○ Ex) 5, 10, 15, 20… 30 � The difference, or d (constant), is always 5. � Geometric Sequence – the terms are found by multiplying each preceding term by a common ratio �The common ratio is the number used to multiply ○ Ex) 2, 4, 8, 16… 64 � The difference, or r (common ratio), is always 2.

Explaining “n” n = index number � This means that whatever n equals, is the placement of an in the sequence. ex) 5, 10, 15, 20, 25 �When n = 1, an = 5 �When n = 3, an = 15 �When n = 5, an = 25

Recursive Formulas*used to find next term �Arithmetic Recursive Formula �an = an– 1 + d ○an = # in the series ○an– 1 = preceding term

Practice Recursive Formulas ex) Given: 10, 12, 14… Find the next term. 1. Use Formula: a 1 = 10 an = an– 1 + d 1. Find d: What is the difference between each term? 2 , so 2 is the common difference, d 2. Plug into formula: an = 14 + 2 = 16 (an– 1 = 14 because it is the preceding term to the next missing term) 3. What are the next 3 terms?

Explicit Formulas *used to calculate any number in the sequence � Arithmetic Explicit Formula �an = a 1 + (n - 1)d We need to know a 1 and d. Then we can find an any value of n! What if n = 100? ? ?

Practice Explicit Formulas � Arithmetic Practice ex) Given: -7, -1, 5, 11… Find n = 25 1. Use formula: an = a 1 + (n - 1)d 2. Plug in values: an = -7 + (25 -1)6 3. Simply: an = -7 + 150 – 6 4. Result: 137

Series = Sum of a Sequence � Arithmetic Summation Formula �This formula calculates the sum of a finite series. n/2 (a 1 + an)

Practice Summation Formulas � Arithmetic Practice ex) Given: 117, 110, 103… 33 sum. Find 1. Use formula: an = a 1 + (n – 1)d 2. Plug in values: an = 117 + (n – 1)7 3. Simplify: an = 124 – 7 n 4. Solve for n: 33 = 124 – 7 n n = 13

Now for some practice!