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Identify the pattern in the sequence as: arithmetic, geometric, or neither.

Arithmetic Sequences. Identify the pattern in the sequence as: arithmetic, geometric, or neither. 7, 11, 15, 19, …. Answer: arithmetic You added to generate each new term. What is the rule used to generate new terms in the sequence?

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Identify the pattern in the sequence as: arithmetic, geometric, or neither.

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  1. Arithmetic Sequences Identify the pattern in the sequence as: arithmetic, geometric, or neither. 7, 11, 15, 19, … Answer: arithmetic You added to generate each new term. • What is the rule used to generate new terms in the sequence? • Write it as a variable expression, and use nto represent the last number given. • 7, 11, 15, 19, … Thecommon difference is 4. Answer: n + 4 (you add 4 ) What are the next 3 terms in the sequence? 7, 11, 15, 19, 23, 27, 31 circle arithmetic geometric neither Rule _______________________ n + (−8) 17 9 1 Ex #2 49, 41, 33, 25 , _____ , _____ , _____ *Yes, this is just subtraction; however, since arithmetic means adding, write is as addition.

  2. Geometric Sequences Identify the pattern in the sequence as: arithmetic, geometric, or neither. 3, 6, 12, 24, … Answer: geometric You multiplied to generate each new term. • What is the rule used to generate new terms in the sequence? • Write it as a variable expression, and use nto represent the last number given. • 3, 6, 12, 24, … Answer: 2n (times 2) The common ratio is 2. What are the next 3 terms in the sequence? 3, 6, 12, 24, 48 , 96 , 192 circle arithmetic geometric neither Rule _______________________ 1 7 3 7 n 21 3 Ex #27203, 1029, 147, _____ , _____ , _____ *Yes, this is just division; however, since geometric means multiply, write is as multiplication.

  3. Other Sequences Describe the pattern in the sequence and identify the sequence as arithmetic, geometric, or neither. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … Answer: neither There’s a pattern, but you’re neither adding nor multiplying by the same number. • What is the rule used to generate new terms in the sequence? • Since the pattern is neither arithmetic nor geometric, you can state the rule in words. • 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … Answer: You add the last 2 terms together to generate each new term) What are the next 3 terms in the sequence? = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 , 89 , 144 , 233 , 377

  4. Negative Number Sequences a. −37, −32, −27, −22, … (…it’s rising slowly … signs not changing …) Arithmetic, Geometric, or Neither? Rule? Common Ratio/Difference? Next 3? arithmetic – you’re adding +5 n + 5 common difference = 5 −17, −12, −7… b. −1, −7, −49, −343, … (…it’s rising quickly … signs not changing …) Arithmetic, Geometric, or Neither? Rule? Common Ratio/Difference? Next 2? geometric –multiplying by 7 7ncommon ratio = 7 −2401, −16,807… c. −99, −103, −107, −111 (…it’s falling slowly … signs not changing …) Arithmetic, Geometric, or Neither? Rule? Common Ratio/Difference? Next 2? arithmetic – adding −4 n + −4 commondifference = −4 −115, −119… d. −4 , 12 , −36 , 108 , … (…it’s rising quickly … signs alternating … ) Arithmetic, Geometric, or Neither? Rule? Common Ratio/Difference? Next 2? geometric –multiplying by −3 −3ncommon ratio = −3 −324, 972…

  5. Decimal Number Sequences a. 0.6 , 1.86 , 5.766 , 17.8746 (… number of decimal places increasing … ) (…divide 1st two terms, then the last 2 … same?) Is it arithmetic, geometric, or neither? What’s the rule? List the next 2 terms. geometric – you’re multiplying by 3.1 3.1n 55.41126, 171.774906 (…divide 1st two terms, then the last 2 … same?) b. 4.7 , 7 , 9.3 , 11.6 , … (…subtract 1st two terms, then the last 2 … same?) Is it arithmetic, geometric, or neither? What’s the rule? List the next 2 terms. arithmetic – you’re adding 2.3 n + 2.3 13.9, 16.2, … (…divide 1st two terms, then the last 2 … same?) c. 4.5, 14.75, 25, 35.25,.. (…subtract 1st two terms, then the last 2 … same?) Is it arithmetic, geometric, or neither? What’s the rule? List the next 2 terms. arithmetic – you’re adding 10.25 n + 10.25 45.5, 55.75, … (…subtract 1st two terms, then the last 2 … same?) d. 1.6, 6.4, 25.6, 102.4,.. (…divide 1st two terms, then the last 2 … same?) Is it arithmetic, geometric, or neither? What’s the rule? List the next 2 terms. geometric – you’re multiplying by 4 4n409.6, 1638.4, …

  6. Fractional Sequences • 1, 1, 13, 3, 8 3 244 ●3 = 3 ● 8 =8 ● 6 = 18 1. Find the Least Common Denominator. 3 ●? =24 ● =24 ● ? =24 6 8 ? , ? , ? , ? , … 8 3 244 • Rewrite each fraction with a new • numerator and denominator. Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms. arithmetic – you’re adding by 5n + 523 , 28 , 33 or 23 , 7 , 11 24 24 24 24 24 24 6 8 28 , 7 , 7 , 7 , … 416 ● 4 = 28 ●16= ● 16 = 448 112 1. Find the Least Common Denominator. 1 1 4 ● ? =16 16 ● =16 ● ? =16 16 ? , ? , ? , ? , … 416 • Rewrite each fraction with a new • numerator and denominator. Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms. geometric – you’re multiplying by 0.25, or 10.25n or 1n7 , 7 , 7 4 4 64 256 512

  7. 0, 4.5, 9, 13.5, … ‒3, ‒ 6, ‒12, ‒24, ‒48, . . . 1, ‒3, 9, ‒27, 81, . . . 1, 2, 1, 2, 1, . . . ‒4, 4, ‒4, 4, ‒4, . . . 0.5, 2.5, 4.5, 6.5, … 7, 4, 1, ‒2, ‒5, . . . ‒5, 10, ‒20, 40, ‒80, . . . Practice with Sequences Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms geometric 2n −96, −192, −384 arithmetic n + 4.5 18, 22.5, 27 Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms neither add 1, then add −1 2, 1, 2 geometric −3n −243, 729, −2187 Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms arithmetic n + 2 8.5, 10.5, 12.5 geometric −1n 4, −4, 4 Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms geometric −2n 160, −320, 640 arithmetic n + (−3) −8, −11, −14

  8. 0, ‒2, ‒5, ‒9, ‒14, . . . 81, 27, 9, 3, 1, . . . ‒80, ‒76, ‒72, ‒68, ‒64, . . . 0.3, 0.6, 0.9, 1.2, … More Practice with Sequences Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms neither add −2, then add −3, then −4, … −20, −27, −35 geometric n Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms arithmetic n + 4 −60, −56, −52 arithmetic n + 0.3 1.5, 1.8, 2.1 Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms geometric n arithmetic n + , 4, Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms arithmetic n + geometric 6n , 16, 96 , , 1

  9. Functions A Coke machine charges $1.00 for a soda. ~ If your input is 1 quarter, your output will be 0 sodas. ~ If your input is 2 quarters, your output will be 0 sodas. ~ If your input is 4 quarters, your output will be 1 soda. ~ Later, you input 4 quarters, but the output is 2 sodas? Is the machine doing its functioncorrectly? 1 0 Is the machine doing its functioncorrectly? 2 0 A relation is a function when: ~ No inputs repeat. or ~ If an input repeats, it’s always paired with the same output. 4 1 4 2

  10. Functions Determine whether the relation is a function. 1. {(–3, –4), (–1, –5), (0, 6), (–3, 9), (2, 7)} Answer: It is NOTa function (an x−value, −3, repeats with a different y−value) 2. {(2, 5), (4, –8), (3, 1), (6, −8), (–7, –9)} Answer: It ISa function (no x−values repeat) 3. 5. 6. 4. It is NOTa function (an x−value, 1, repeats with a different y−value) It ISa function (an x−value, −4, repeats with the SAME y−value, 11) It ISa function (no x−values repeat) It ISa function (no x−values repeat).

  11. Functions Determine whether each graph is a function. Explain. If NO x−values repeat, it IS a function. If NO x−values repeat, it IS a function. Use “vertical line test” to test for a function: Use “vertical line test” to test for a function: 1. Hold a pencil vertically ... 1. Hold a pencil vertically ... 2. Then, slide it across the curve. 2. Then, slide it across the curve. *Does the pencil ever hit the curve TWICE? *Does the pencil ever hit the curve TWICE? If the pencil hits the curve ONCE, it IS a function. If the pencil hits the curve TWICE, it is NOT a function. If the pencil hits the curve ONCE, it IS a function. If the pencil hits the curve TWICE, it is NOT a function. The pencil hits the curve ONCE, so it PASSES the vertical line test. It IS a function. The pencil hits the curve TWICE, so it FAILS the vertical line test. It is NOT a function.

  12. Functions There are different ways to show each part of a function. Let’s use the example of: The effect of temperature on cricket chirps This is the list of all input (x) values. This is the list of all output (y) values. Which variable causes the change? Which variable responds to the change? Which letter is listed first in an ordered pair? Which letter is listed second in an ordered pair? This is what goes in. This is what comes out. Conclusion: As temperature increases, cricket chirps increase. (Summary):

  13. Functions A teacher displays the results of her survey of her students. texts per week average quiz score a. What is the input? output? texts per week avg quiz score b. What is the independent variable? dependent variable? {10, 25, 100, 200} {81, 87, 94} c. What are all the x−values? y−values? {10, 25, 100, 200} {81, 87, 94} d. What’s the domain? range?

  14. Writing Functions As Variable Expressions Matt is a manager at Dominos. He earns a salary of $500/week, but he also gets $0.75 for every pizza he sells. Write a variable expression you could use to find his total weekly pay. Ned sells tandem skydives. He makes $1000 for a full plane of jumpers, but he has to pay the pilot $25 per jumper. Write a variable expression you could use to find his total pay for every full plane. salary + pay per pizza = total weekly pay Ned’s pay ‒ pay per jumper = total pay 500 + 0.75 • p 1000 ‒ 25 • j 500 + 0.75p 1000 ‒25j Lambert is running a food donation drive, and the results are to the right. Write a variable expression you could use to find his total pounds of food donated? Adam drives a truck, and his mileage chart is above. Write a variable expression you could use to find his total amount of gas he has in his tank? starting food + food per day 97 + 2 • d gas he started with – gas per mile = gas remaining 35.1 − 0.6 • m + d 35.1 ‒0.6m

  15. Completing a Function Table To graph a function ~ Step 1: Pick a value for x ( I recommend “0”), then ... * Write “0” under “x”, ... * ... re−write your equation, then plug in “0” for x, then ... * ... plug in “0” for the x−value of the ordered pair. ~ Step 2: To figure out the y−value, * Use order of operations to evaluate the expression. The “answer” is your y−value, so ... > write it under “y”, ... > ... then plug it in for the y−value of your ordered pair. xy = 4x + 3 y (x,y) y = 4( ) + 3 ( , ) y = 4( ) + 3 y = 4( ) + 3 3 0 3 0 0 ( , ) 1 7 1 1 7 11 ( , ) 2 11 2 2

  16. Graphing Functions with Ordered Pairs Plot all three ordered pairs from your function table If they all line up, ~ get a ruler, then ... ~ draw a straight line through all 3 points. If they don’t line up, ~ choose a new x−value ~ plug it in your function table ~ plot your new point (hopefully, they line up)

  17. Completing a Function Table To graph a function ~ Step 1: Pick a value for x ( I recommend “0”), then ... * Write “0” under “x”, ... * ... re−write your equation, then plug in “0” for x, then ... * ... plug in “0” for the x−value of the ordered pair. ~ Step 2: To figure out the y−value, * Use order of operations to evaluate the expression. The “answer” is your y−value, so ... > write it under “y”, ... > ... then plug it in for the y−value of your ordered pair. xy = x – 2y(x,y) y = ( ) – 2 y = ( ) – 2 y = ( ) – 2 1 4 ( , ) 0 0 –2 –2 1 4 0 ( , ) 1 4 4 –1 4 –1 4 1 4 0 8 8 8 0 ( , )

  18. Graphing Functions with Ordered Pairs Plot all three ordered pairs from your function table If they all line up, ~ get a ruler, then ... ~ draw a straight line through all 3 points. If they don’t line up, ~ choose a new x−value ~ plug it in your function table ~ plot your new point (hopefully, they line up)

  19. Graphing Horizontal (y =) Lines. Graph y = 4 ~ Write an ordered pair with any x−value. ( 0 , ) 4 ~ The y−value is 4. * Why? Because the original equation is y = 4. ~ Pick another x−value. The y−value will be 4. ( , 4 ) 1 2 ( , 4 ) ~ Plot the points, then draw your line.

  20. Graphing Vertical (x = ) Lines. Graph x = –7 ~ Write an ordered pair with any y−value. ( , 0 ) –7 ~ The x−value is –7. * Why? Because the original equation is x = –7. ~ Pick another y−value. The x−value is –7. (–7 , ) 1 (–7 , ) 2 ~ Plot the points, then draw your line.

  21. In an arithmetic sequence, you add the ________ to get each new term. 14, 3, −8, ... +(−11) +(−11) common difference

  22. In a geometric sequence, you multiply by the ________ to get each new term. 3, 21, 147,... •7 •7 common ratio

  23. 6n , geometric n + 12 , arithmetic 7n , geometric 3n , geometric •3, •4, •5 … , neither n + 1.1 , arithmetic 20.6, 24.6, 28.6 204.8, 1638.4, 13107.2 53, 58, 63 768, 3072, 12288 1.3, 1.6, 1.9 Sequences 2013, 2020, 2037

  24. Arithmetic Sequences Identify the pattern in the sequence as: arithmetic, geometric, or neither. 7, 11, 15, 19, … Answer: arithmetic You added to generate each new term. • What is the rule used to generate new terms in the sequence? • Write it as a variable expression, and use nto represent the last number given. • 7, 11, 15, 19, … Answer: n + 4 (since you add 4 to generate each new term) What are the next 3 terms in the sequence? 7, 11, 15, 19, 23, 27, 31 circle arithmetic geometric neither Rule _______________________ n + (−8) 17 9 1 Ex #2 49, 41, 33, 25 , _____ , _____ , _____ *Yes, this is just subtraction; however, since arithmetic means adding, write is as addition.

  25. Geometric Sequences Identify the pattern in the sequence as: arithmetic, geometric, or neither. 3, 6, 12, 24, … Answer: geometric You multiplied to generate each new term. • What is the rule used to generate new terms in the sequence? • Write it as a variable expression, and use nto represent the last number given. • 3, 6, 12, 24, … Answer: 2n(since you multiplied by 2 to generate each new term) What are the next 3 terms in the sequence? 3, 6, 12, 24, 48 , 96 , 192 circle arithmetic geometric neither Rule _______________________ 1 7 3 7 n 21 3 Ex #27203, 1029, 147, _____ , _____ , _____ *Yes, this is just division; however, since geometric means multiply, write is as multiplication.

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