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### BHS Math Survival Kit

### Math 3 Module 4: Rational Functions

### 1.1 Ready, Set, Go!

### Questions 1 - 4

### You're given three possible options for x and need to figure out which one is right. You could try putting each value of x into the expression and figure out which x makes the expression true. This video shows you how to solve for x in a similar linear equation: https://www.youtube.com/watch?v=wShnYemIr28

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### Question 5

### In this problem, you are given an expression and three "ordered pairs" or "coordinates" which look like: (x,y). However, you are only given the x value for each. For the first ordered pair, take the value x = 8 and find the value of y in the expression y = 6x − 15. Your answer is the y value for the first ordered pair. Do the same for the other two!

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### Question 6

### In this problem, you are given an expression and three "ordered pairs" or "coordinates" which look like: (x,y). However, you are only given the x value for each. For the first ordered pair, take the value x = -5 and find the value of y in the expression y = −4x + 9;. Your answer is the y value for the first ordered pair. Do the same for the other two!

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### Question 7

### In this problem, you are given an expression and three "ordered pairs" or "coordinates" which look like: (x,y). However, you are only given the x value for each. For the first ordered pair, take the value x = -4 and find the value of y in the expression y = 2x − 1. Your answer is the y value for the first ordered pair. Do the same for the other two!

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### Question 8

### In this problem, you are given an expression and three "ordered pairs" or "coordinates" which look like: (x,y). However, you are only given the x value for each. For the first ordered pair, take the value x = -9 and find the value of y in the expression y = −x + 9;. Your answer is the y value for the first ordered pair. Do the same for the other two!

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### Be careful about making arithmetic errors when finding your y-value, especially with negative x-values, remember that a negative number minus a negative number is like adding two negatives numbers. (Ex. = (-8) - 5 = (-8) + (-5))

### Question 9

### To set up your business, you've already spent $600 and when you make a birdcage to sell, it costs you another $5. After that, each additional birdcage is $5. Fill out the table.

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### Question 10

### In this question you start with a $500 loan and pay back $15 on day 1. Each additional day later, you take off another $15. Fill out the table.

### Question 11

### Your savings account starts at $0 and you add $10 on week 1. Each week after that, you add another $10. Fill out the table.

### Question 12

### You have $25 already and add $10 after the 1st week. Each additional week you add another $10. Fill out the table.

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### Questions 13 - 16

### In this question you have to plot the "ordered pairs" (also known as "coordinates") provided in the table. This video goes over how to plot ordered pairs on a cartesian plane: https://www.youtube.com/watch?v=j6LGxJhc8Kk

### For the table of ordered pairs that you're given, the points should form a line like in this video: https://www.youtube.com/watch?v=-u55GD_sGLA

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### 1.2 Ready, Set, Go!

### Questions 1 - 8

### In this question you need to evaluate the function. This video gives you some similar examples: https://www.youtube.com/watch?v=pJjcqlGUPg4

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### Questions 9 - 12

### You need to figure out what the pattern is and then add the missing entries, like this video shows: https://www.youtube.com/watch?v=1Ww6U__uy6A

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### Question 13

### All you need to do is draw the next figures in the pattern. This video shows a similar pattern: https://www.youtube.com/watch?v=No4VLQLrdr0

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### Question 14

### Look at expression (a), (b) and (c) and decide which one fits the scenario.

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### Question 15

### If "n" is the number of tiles and "s" is the number of steps, then number of tiles "n" is equal to 3 times "s" minus 2.

### Question 16

### Add up the numbers in the right column in the table to start with. Then the easiest way to do this is to set s = 1, 2, 3, ... and calculate n for each expression and then compare it to the values in the table and see which expression matches.

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### I also noticed that one of the expressions had a formula for odd numbers in it (2s - 1) and then adds the number of tiles above the base which is 0 when s=1, and is 1 when s=2.

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### Questions 17 - 24

### You need to turn this multiplication expression into an exponent expression. This video reviews exponents: https://www.youtube.com/watch?v=XZRQhkii0h0

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### Questions 25 - 28

### In this problem you are multiplying an exponent by a whole number. This video provides similar examples: https://www.youtube.com/watch?v=BVwFp0-pmRE

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### 1.3 Ready, Set, Go!

### Question 1

### For part (A) look along the table for f(n) = 12 and find the value for n above the 12.

### For part (B), you need to evaluate f(n - 1). Use the value that you found for n in part (A). For example, if you found n = 101, then f(n - 1) would be f(100) (which is outside of the table)

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### Question 2

### For part (A) look along the table for f(n) = 17 and find the value for n above the 17.

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### For part (B), you need to evaluate f(n - 1). Use the value that you found for n in part (A). For example, if you found n = 101, then f(n - 1) would be f(100) (which is outside of the table)

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### Question 3

### For part (A) look along the table for f(n) = 32 and find the value for n above the 32.

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### For part (B), you need to evaluate f(n - 1). Use the value that you found for n in part (A). For example, if you found n = 101, then f(n - 1) would be f(100) (which is outside of the table)

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### Question 4

### For part (A) look along the table for f(n) = 2 and find the value for n above the 2.

### For part (B), you need to evaluate f(n + 3). Use the value that you found for n in part (A). For example, if you found n = 3, then f(n + 3) would be f(3+3) or f(6). Which means, when n is 6, what is f(n)?

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### Question 5

### For part (A) look along the table for f(n) = 27 and find the value for n above the 27.

### For part (B), you need to evaluate f(n - 6). Use the value that you found for n in part (A). For example, if you found n = 8, then f(n - 6) would be f(8-6) or f(2). Which means, when n is 2, what is f(n)?

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### Question 6

### For part (A) look along the table for f(n) = -8 and find the value for n above the -8.

### For part (B), you need to evaluate f(n + 9). Use the value that you found for n in part (A). For example, if you found n = 1, then f(n + 9) would be f(1+9) or f(10). Which means, when n is 10, what is f(n)?

### Question 7

### Set x = 4 in the explicit equation and evaluate to get the value of the 4th term.

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### If you want to use the recursive formula, then this video is helpful: https://www.youtube.com/watch?v=RjsyEWDEQe0

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### Question 8

### Set x = 50 in the explicit equation and evaluate to get the value of the 50th term.

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### If you want to use the recursive formula, then this video is helpful: https://www.youtube.com/watch?v=RjsyEWDEQe0

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### Question 9

### Set x = 20 and evaluate the explicit equation to find the value of the 20th term.

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### If you want to use the recursive formula, then this video is helpful: https://www.youtube.com/watch?v=RjsyEWDEQe0

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### Question 10

### Set x = 9 and evaluate the explicit equation to find the value of the 20th term.

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### Question 11

### Set x = 5 and evaluate the explicit equation to find the value of the 20th term.

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### Question 12

### Set x = 7 and evaluate the explicit equation to find the value of the 20th term.

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### Questions 13 - 17

### In this question you have to evaluate the exponential and fill the table, like the example in this video: https://www.youtube.com/watch?v=PEtIQqvIoGU

### 1.4 Ready, Set, Go!

### Questions 1 - 8

### In this question you need to evaluate the function at the value you are given. This video gives you some similar examples: https://www.youtube.com/watch?v=pJjcqlGUPg4

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### Questions 9 - 12

### For part (A) and (B) you need to figure out the pattern and finish the sequence, like in this video: https://www.youtube.com/watch?v=EadlY9iuIIM

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### In part (C) you need to give the recursive and explicit functions that describes the sequence, like this: https://www.youtube.com/watch?v=1W2Ziv4m0Bo (IMPORTANT NOTE: in this video, they use f(n) to describe "now" and f(n-1) to describe "previous")

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### Questions 13 - 14

### If you need help reading coordinates off graphs, this video might be a good refresher: https://www.youtube.com/watch?v=j6LGxJhc8Kk

### Question 15

### Look at Olaf's original elevation at noon and subtract his elevation at 2pm to find the distance has has walked down the mountain. If you need help reading coordinates off graphs, this video might be a good refresher: https://www.youtube.com/watch?v=j6LGxJhc8Kk

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### Question 16

### Functions such as f(n) usually have inputs such as time and distance. Based on the graph you're given, the function is probably calculating the elevation at time n.

### 1.5 Ready, Set, Go!

### Questions 1 - 4

### To find the rate of change in a table, check out this video: https://www.youtube.com/watch?v=wasEKQkA7MA

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### For help finding the rate of change on a graph, take a look at this other video: https://www.youtube.com/watch?v=MYCH7gswI4k

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### Questions 5 - 6

### This video explains how to find the explicit and recursive formulas for geometric sequences: https://www.youtube.com/watch?v=8a1a5A3CfdQ

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### If you want a refresher on graphing, this video shows you what you can expect arithmetic and geometric sequences to look like when they are graphed: https://www.youtube.com/watch?v=jU6JTdhHVuI

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### Question 7

### Use the information given to create a table with 4 rows (like the table in question 6) and fill the right column with the amount of money left in Claire's account each month. After 1 month, she takes out half the money and is left with $150. After 2 months, she takes out half of what was in there from the previous month. Calculate how much she will have after 2, 3, and 4 months to fill out the table. From there, you can find the recursive and explicit formulas for geometric sequences like this: https://www.youtube.com/watch?v=8a1a5A3CfdQ

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### If you want a refresher on graphing, this video shows you what you can expect arithmetic and geometric sequences to look like when they are graphed: https://www.youtube.com/watch?v=jU6JTdhHVuI

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### Question 8

### Use the information given to create a table with 4 rows (like the table in question 6) and fill out the right column with the number of people that will receive the chain letter each time. For example, on day 1 Tania sends the letter to 4 friends, so 4 people have received it. On day 2, those 4 friends send the letter to another 4 friends each, so 16 people receive the letter that day. Figure out how many people receive letters on days 3 and 4! From there, you can find the recursive and explicit formulas for geometric sequences like this: https://www.youtube.com/watch?v=8a1a5A3CfdQ

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### If you want a refresher on graphing, this video shows you what you can expect arithmetic and geometric sequences to look like when they are graphed: https://www.youtube.com/watch?v=jU6JTdhHVuI

### Question 9

### Count up the number of dots in each sequential figure and put it into a table. From there, you can find the recursive and explicit formulas for geometric sequences like this: https://www.youtube.com/watch?v=8a1a5A3CfdQ.

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### Questions 10 - 13

### This video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

### This one goes through finding the explicit formula: https://www.youtube.com/watch?v=T1ZBizIL5Lw

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### Question 14

### Count up the number of dots in each sequential figure and put it into a table. From there, you can find the recursive and explicit formulas like this: https://www.youtube.com/watch?v=8a1a5A3CfdQ

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### 1.6 Ready, Set, Go!

### Questions 1 - 6

### This video shows you how to find the missing terms in an arithmetic sequence: https://www.youtube.com/watch?v=g7Tl3clSDYw

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### Questions 7 - 9

### I found a video that shows you how to find the recursive formula of an arithmetic sequence given two terms. The example uses different notation but don't let that scare you! His aâ‚ƒ is the same as f(3)). https://www.youtube.com/watch?v=VguyPrGaXLk

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### Questions 10 - 12

### To find the explicit form of the geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

### When you have explicit formula, you can convert it to a recursive formula like this: https://www.youtube.com/watch?v=Iq7a2vEsT-o

### Questions 13 - 16

### This question asks you to evaluate the exponential function they've given you. This video shows you how to evaluate exponential functions: https://www.youtube.com/watch?v=GQIcZeKR9RI

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### Question 17

### Look at the parentheses in question 15, how do they change the effect of the exponent? https://www.youtube.com/watch?v=pJjcqlGUPg4

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### Questions 18 - 19

### These questions asks you to evaluate the linear function they've given you. This video shows you how to evaluate linear functions: https://www.youtube.com/watch?v=GQIcZeKR9RI

### 1.7 Ready, Set, Go!

### Questions 1 - 4

### Look at the pattern and fill in the missing numbers. This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

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### Questions 5 - 8

### Look at the pattern to decide if the sequence is arithmetic or geometric. This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

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### This video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

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### This one goes through finding the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

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### Question 9

### Use the story to write out the first 4 or 5 numbers in the sequence. Look at the pattern to decide if the sequence is arithmetic or geometric. This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

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### To find the explicit form of the geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

### When you have explicit formula, you can convert it to a recursive formula like this: https://www.youtube.com/watch?v=Iq7a2vEsT-o

### Questions 10 - 12

### This explains the difference between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

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### This video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

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### This one goes through finding the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

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### To find the explicit form of the geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

### When you have explicit formula, you can convert it to a recursive formula like this: https://www.youtube.com/watch?v=Iq7a2vEsT-o

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### Questions 13 - 18

### This video shows you how to find slope of the line that runs through two points: https://www.youtube.com/watch?v=8trWFtwyUMU

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### If you need a refresher on plotting coordinates on a graph, here's a good video for that: https://www.youtube.com/watch?v=s7NKLWXkEEE

### 1.8 Ready, Set, Go!

### Questions 1 - 4

### This video shows you how to find the common ratio for a geometric sequence: https://youtu.be/-rofZ1OtTik

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### Questions 5 - 8

### Start by looking at the pattern and decide if the sequence is arithmetic or geometric. This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

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### This one goes through finding the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

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### Questions 9 - 12

### You need to find the equation of the line on the graph. This video has an example of a line with a positive slope: https://www.youtube.com/watch?v=mmWf_oLTNSQ

### And this video is an example of a line with a negative slope: https://www.youtube.com/watch?v=dBFbFMvEwQg

### 1.9 Ready, Set, Go!

### Questions 1 - 2

### This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

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### Questions 3 - 6

### This video has examples of finding missing terms in an arithmetic sequence: https://www.youtube.com/watch?v=g7Tl3clSDYw

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### This video shows you how to find the formula for the n-th term in an arithmetic sequence:

### https://www.youtube.com/watch?v=lj_X9JVSF8k

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### Questions 7 - 11

### Start by looking at the pattern and decide if the sequence is arithmetic or geometric. This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

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### 1.10 Ready, Set, Go!

### Questions 1 - 5

### You are given the first term and common difference, d, for an arithmetic sequence f(d). You need to find the first 5 terms of the sequence with that information. This video shows you how to do a similar question - notice that you already have the common difference, so you don't need to calculate it: https://www.youtube.com/watch?v=eR_kQf-1fks

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### You are given the first term and common ratio, r, for a geometric sequence g(r). You need to find the first 5 terms of the sequence with that information. This video shows you how to do a similar question - notice that you already have the common difference, so you don't need to calculate it: https://www.youtube.com/watch?v=3uiTQ6JSMC0

### Question 6

### Arithmetic sequences are straight lines on a graph and geometric sequences are exponential curves. This video compares linear and exponential curves - you can decide if exponential curves always outgrow lines! https://www.youtube.com/watch?v=TVX9pSPJwYA

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### Questions 7 - 10

### This video has examples of finding missing terms in an arithmetic sequence: https://www.youtube.com/watch?v=tqli6fOnph8

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### This video shows you how to find the formula for the n-th term in an geometric sequence:

### https://www.youtube.com/watch?v=IGFQXInm-co

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### Question 11

### You are given the first term and common ratio, r, for an geometric sequence g(r). You need to find the first 5 terms of the sequence with that information. This video shows you how to do a similar question - notice that you already have the common difference, so you don't need to calculate it: https://www.youtube.com/watch?v=3uiTQ6JSMC0

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### Questions 12 - 13

### To find the explicit formula for a geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

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### Question 14

### Plug the value x = 100 into the explicit expressions you found in questions 11., 12., and 13. and see which value is largest!

### 1.11 Ready, Set, Go!

### Question 1

### This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

### Khan Academy has a good explanation of arithmetic sequences: https://www.youtube.com/watch?v=_cooC3yG_p0

### Question 2

### "Create a context" means "come up with a story that this sequence would apply to. You can come up with any story that fits the data!

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### If you need a refresher on plotting coordinates on a graph, here's a good video for that: https://www.youtube.com/watch?v=s7NKLWXkEEE

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### Question 3

### In this question you are given the recursive equation for the sequence. I would use this formula to list out the first 4 or 5 points in the table like this: https://www.youtube.com/watch?v=IFHZQ6MaG6w

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### When you've filled the table, this video shows you how to find the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

### Question 4

### In this question you are given the explicit equation for the sequence. Evaluate the function f(n) when n = 1, n = 2, ..., n = 5.

### When you've filled the table, this video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

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### Question 5

### When you've filled the table, this video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

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### Questions 6 - 8

### In this question you are given the recursive equation for the sequence. I would use this formula to list out the first 4 or 5 points in the table like this: https://www.youtube.com/watch?v=IFHZQ6MaG6w

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### When you've filled the table, this video shows you how to find the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

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