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” ex) 5, 10, 15, 20, 25 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 Arithmetic Recursive Formula an = an–1 + d *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. Use Formula: a1 = 10 an = an–1 + d Find d: What is the difference between each term? 2 , so 2 is the common difference, d Plug into formula: an = 14 + 2 = 16 (an–1 = 14 because it is the preceding term to the next missing term) What are the next 3 terms?

Explicit Formulas Arithmetic Explicit Formula an = a1 + (n - 1)d *used to calculate any number in the sequence Arithmetic Explicit Formula an = a1 + (n - 1)d We need to know a1 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 Use formula: an = a1 + (n - 1)d Plug in values: an = -7 + (25-1)6 Simply: an = -7 + 150 – 6 Result: 137

Series = Sum of a Sequence Arithmetic Summation Formula This formula calculates the sum of a finite series. n/2 (a1 + an)

Practice Summation Formulas Arithmetic Practice ex) Given: 117, 110, 103… 33 Find sum. Use formula: an = a1 + (n – 1)d Plug in values: an = 117 + (n – 1)7 Simplify: an = 124 – 7n Solve for n: 33 = 124 – 7n  n = 13

Now for some practice!