Sequences. An _arithmetic sequence_ is a sequence in which the difference between any two consecutive terms is the same.

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1 Name: Unit3 #7 Sequences A sequence is an ordered list of numbers. Each number is called a _term of the sequence. An _arithmetic sequence_ is a sequence in which the difference between any two consecutive terms is the same. Arithmetic sequence each term is found by _adding the same number to the previous term 8, 11, 14, 17, "n + 3" The difference between any two consecutive terms in an arithmetic sequence is called the _common difference. State whether the sequence 3, 5, 7, 9, 11, is arithmetic. If it is, state the common difference and write the next three terms. 3, 5, 7, 9, 11, _13_, _15_, _17_, Notice, that 5 3 = = = 2, and so on. Write the rule as an algebraic expression: The previous term n, increased by 2 n + 2 Sometimes the arithmetic sequence looks like the term is subtracted, but it s really adding a negative. State whether the sequence is arithmetic. If it is, find the values of the following: 17, 12, 7, 2, 3, 4, 1, 2, 5, 20, 16, 12, Common difference: _ 5 Common difference: _ 3 Common difference: Next three terms: 8, 13, 18, Next three terms: _ 8, 11_, 14_ Next three terms:,, Ninth term: 23 Eighth term: _ 17 Seventh term: Rule: _n + ( 5) Rule: _n + ( 3) Rule: If the numbers are growing, then the common difference is positive. If the numbers are getting smaller, then the common difference is negative.

2 Geometric sequence each term is found by multiplying term by the same number 3, 6, 12, 24, "2n" The factor between any two consecutive terms in a geometric sequence is called the _common factor. State whether the sequence 2, 4, 8, 16, 32, is geometric. If it is, state the common ratio and write the next three terms. 2, 4, 8, 16, 32, _ 64_, _ 128_, _ 256_, Notice, that = 2 = 2 = 2, and so on. Write the rule as an algebraic expression: The previous term n, multiplied by 2 _n 2 OR 2n_ Sometimes the geometric sequence looks like the term is divided, but it s really multiplying a fraction. State whether the sequence is geometic. If it is, find the values of the following: 81, 27, 9, 3, 1,, 1, 5, 25, 125, 32, 8, 2, Common ratio: Common ratio: _ 5 Common ratio: Next three terms:,, Next three terms: 625, 3125, Next three terms:,, Ninth term: Eighth term: Seventh term: Rule: Rule: _ 5n Rule: If the signs of all terms are the same (all positive OR all negative), then the common ratio is positive. If the sign change for every next term, then the common ratio is negative.

3 State if each sequence is arithmetic, geometric, or neither. If the sequence is arithmetic or geometric, fill in the table and describe the relationship (write the rule) between two consecutive terms in the sequence. Sequence arithmetic, Common Next three terms Relationship geometric, or neither difference/ratio 0, 3, 6, 9, 12,... arithmetic 3 15,18, 21 n , 24, 12, 6, 3,... geometric,, 6, 11, 16, 21, 26,... arithmetic 5 31, 36, 41 n + 5 0, 1, 3, 6, 10,... neither none 15, 21, 28, 1, 3, 9,... geometric 3 27, 81, 243 3n 30, 26, 22, 18, 14,... arithmetic 4 10, 6, 2 n 4 3, 6, 12, 24,... geometric 2 48, 96, 192 2n 4, 4, 4, 4, 4,... geometric 1 4, 4, 4 1 n 5, 10, 20, 40,... geometric 2 80, 160, 320 2n 1, 2, 1, 2, 1,... neither none 2, 1, 2 448, 224, 112, 56,... geometric 28, 14, 7 1. Write a sequence for the perimeters of the rectangles. 12, 16, 20, 24 Is the sequence arithmetic or geometric? arithmetic Common difference/ratio: 4 Next three terms: _28_, 32, 36 Rule: n Write a sequence for the areas of the rectangles: 8, 15, 24, 35 Is the sequence arithmetic or geometric? neither Common difference/ratio: none Next three terms: 48, 63, 80 Rule: add 2 more than we added the previous time 3. Write an arithmetic sequence in which the common difference is 5.

4 HOMEWORK 3.7 Sequence 35, 28, 21, 14 arithmetic, geometric, or neither Common difference/ratio Next three terms Relationship 1, 3, 9, 27 2, 4, 8, , 1221, 12221, 1, 2, 4,... 8, 6, 4, 2 2, 6, 24, 120 2, 6, 18, 54 8) What is the 7th term of the arithmetic sequence: 29, 21, 13, 5, 9) Find the missing terms in the sequence 26, 33, 40,, 54, 61,, 13) Create three sequences. One sequence must be arithmetic, one sequence must be geometric, and one sequence must be neither. Then describe the relationship between two consecutive terms in each of your arithmetic and geometric sequences. Arithmetic: Geometric: Neither: Look for a pattern. Then use the pattern to solve each problem. There were 256 people at a fundraiser. When the event was over, half of the people who remained left every 5 minutes. How long after the event ended did the last person leave? A knitting shop is having a huge yarn sale. One skein sells for $1.00, 2 skeins sell for $1.50, and 3 skeins sell for $2.00. If this pattern continues, how many skeins of yarn can you buy for $5.00?

5 The football cheerleaders will arrange themselves in rows to form a pattern on the football field at halftime. In the first five rows there are 12, 10, 11, 9, and 10 girls in each row. They will form a total of twelve rows. If the pattern continues, how many girls will be in the back row? Find the perimeters of the next two figures in the pattern. The length of each side of each small square is 3 feet. A hot tub holds 630 gallons of water when it is full. A hose fills the tub at a rate of 6 gallons every five minutes. How long will it take to fill the hot tub? Jordan saved $1 the first week, $2 the second week, $4 the third week, and $8 the fourth week. If this pattern continues, how much will she save the eighth week?