Do Now Find the pattern and use it to find the next 3 numbers in the sequence: 1, 3, 5, 7, 9, 11,___, ___, ___ 2, 7, 12, 17, 22, 27,___, ___, ___ 100, 97, 94, 91, 88, 85,___, ___, __

Arithmetic and Geometric Sequences

Arithmetic Sequences Every day a radio station asks a question for a prize of $150. If the 5th caller does not answer correctly, the prize money increased by $150 each day until someone correctly answers their question.

Arithmetic Sequences Make a list of the prize amounts for a week (Mon - Fri) if the contest starts on Monday and no one answers correctly all week.

Arithmetic Sequences Monday : $150 Tuesday: $300 Wednesday: $450 Thursday: $600 Friday: $750

Arithmetic Sequences These prize amounts form a sequence, more specifically each amount is a term in an arithmetic sequence. To find the next term we just add $150.

What is an arithmetic sequence? A sequence in which each term is found by ADDING the same number to the previous term. 4, 8, 12 , 16, 20………….. +4 +4 +4 +4

What is the common difference? The difference between each number. This determines what is added to each previous number to obtain the next number. 4, 8, 12, 16, 20………….. 4 is the common difference

Arithmetic Sequences An arithmetic sequence is a set of numbers put into a specific order by a pattern of addition or subtraction. an = a1 + (n – 1)d– This is the formula. an represents the nth term, the unknown term that you are trying to find, of a sequence. a1 is the first term in a sequence. n is an unknown term that is always the same number as the n term in an.

Arithmetic Sequences (continued) an=a1+(n-1)d The d in the formula is the Common Difference between each of the terms in a series. For example: 1, 5, 9, 13… The common difference (d) is +4. The d term can also be negative: 10, 7, 4, 1, -2… The d term is -3 (this means that instead of adding a number you subtract it.)

Try These! Are they arithmetic sequences? Find the pattern and use it to find the next 3 numbers in the sequence: 1, 3, 5, 7, 9, 11,___, ___, ___ 2, 7, 12, 17, 22, 27,___, ___, ___ 100, 97, 94, 91, 88, 85,___, ___, __

What is a geometric sequence? A sequence in which each term can be found by multiplying the previous term by the same number. 3, 9, 27 , 81, 243………….. x3 x3 x3 x3

What is the common ratio? The number used to multiply by each previous number to obtain the next number. 2, 8, 32, 128, 512………….. 4 is the common ratio

Geometric Sequence What if your pay check started at $100 a week and doubled every week. What would your salary be after four weeks?

GeometricSequence Starting $100. After one week - $200 After two weeks - $400 After three weeks - $800 After four weeks - $1600. These values form a geometric sequence.

Geometric Sequences an = a1rn-1 Geometric Sequence formula. an is the unknown term (just like the arithmetic sequences) a1 is the first term. r is the rate, also known as the common ratio. It is the change between two terms in a geometric sequence. It is either a number being multiplied or divided. You can also multiply by (1) over the number being multiplied.

More Geometric Sequences Some examples of geometric sequences are: 1, 2, 4, 8, 16, 32…-- r = 2 100, 50, 25, 12.5, 6.25…-- r = 1/2 (divide the preceding number by 2.) an=a1rn-1

How this relates to Real Life Outside Math Class A painter is a job that requires the use of an arithmetic sequence to correctly space the things he is painting. If the painter was painting stripes on a wall, he could find the places to put the stripes to evenly space them.

Another Real Life Slide If an owner of a store needed to count up the amount of stuff they sell, or how much money they make, he could use and arithmetic or geometric sequence. If the owner had a pattern of how much money they make as time progresses, that is a sequence. The owner also needs these sequences if he/she wants to predict the earnings of his or her store in years to come.

Let’s Practice!

Is this an arithmetic or geometric sequence? 10, 15, 20, 25, 30…… Arithmetic Sequence +5 5 is the common difference

Is this an arithmetic or geometric sequence? 2, 12, 72, 432, 2,592…… Geometric Sequence x6 6 is the common ratio

What is the next term in this sequence? 5, -1, -7 -13 , _____ Is this an arithmetic or geometric sequence? What is the common ratio or difference? What is the next term in this sequence? 5, -1, -7 -13 , _____ -6 -6 -6 -6 -19

What is the next term in this sequence? -400,-380,-360,-340 , ____ Is this an arithmetic or geometric sequence? What is the common ratio or difference? What is the next term in this sequence? -400,-380,-360,-340 , ____ +20 +20 +20 +20 -320

What is the next term in this sequence? 12, -48, 192 , -768____ Is this an arithmetic or geometric sequence? What is the common ratio or difference? What is the next term in this sequence? 12, -48, 192 , -768____ x-4 x-4 x-4 x-4 3,072

Some Interesting Example Equations Geometric example: find the nth term. a1 = -10, r=4, n=2 an = -10(4)2-1 an = -10(4)1 an = -40 Arithmetic example: find a14, a1=4, d=6 a14= 4 + (14-1)6 a14= 4 + 78 a14= 82