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1. Use the distributive property to solve the following problem: Remember to combine like terms. 4 + 6(n -1) 3, 9, 27, 81,… What would the 5th number be, the 10th number, the 73rd number? Activator

2. The nth term What is it? The nth term is used to find any unknown number in a pattern. For example. What is the 50th number in this pattern? 2, 5, 8, 11, 14 … 1st term 2nd term Do you really have time to figure this out? Nah.

3. A sequence is an ordered list of numbers. Each number in a sequence is called a term. When the sequence follows a pattern, the terms in the sequence are the output values of a function, and the value of each number depends on the number’s place in the list.

4. By "the nth term" of a sequence we mean an expression that will allow us to calculate the term that is in the nth position of the sequence. For example consider the sequence • 2, 4, 6, 8, 10,... • The pattern is easy to see. We are adding 2 to the previous number. • The first term is two. • The second term is 4. • The third term is 6 • The fourth term is 8 • The tenth term is … • the nineteenth term is ….

5. Arithmetic Sequence: When the value changes by a common number added or subtracted. The difference is called the common difference. There are two types of patterns

6. When to use: Equation: Arithmetic nth term rules

7. When to use: Use when the pattern is increasing or decreasing by addition or subtraction. Equation: A + d(n-1) A = 1st term in pattern D = the difference between each term N = the position in the pattern you are trying to find. Arithmetic nth term rules

8. n (position in the sequence)‏ 1 2 3 4 y (value of term)‏ 2 4 6 8 You can use a variable such as n, to represent a number’s position in a sequence.

9. The nth term of a linear number sequence (a sequence that goes up or down by the same amount each time) can be found by using the following mathematical formula: nth term = a + d(n-1) • a is the first number in the number sequence • d is the common difference (what you are adding or taking from term to term). • n is the number of the term you are trying to find.

11. Step 3 nth term formula is a + d(n-1)

13. Find the Nth term rule for each sequence To write the nth term rule always start with a+d(n-1) find the common difference then simplify your expression. This is the rule of your sequence.

14. 1. 6, 12, 18, 24,… • 2. 24, 21, 18, 15,… • 3. -28, -24, -20… • 4. 45, 42, 39… Find the nth term expression

15. 1. 1, 7, 13, 19, … Find the 25th term 2. 3, 6, 9, 12, … Find the 10th term 3. 18, 14, 10, 6 … find the 30th term 4. -55, -50, -45, -40,… find 100th term Find the nth term rule and then the missing numbers.

16. Assessment Prompt How is finding the nth term different than finding a function rule?

17. From yesterday, • What does a, d, and n stand for? Activator

18. In ageometric sequence, each term is multiplied by the same amount to get the next term in the sequence. There are two types of patterns

19. n (position in the sequence)‏ 1 2 3 4 y (value of term)‏ 2 6 18 54 • A sequence where the numbers increase by multiplication. • Formula: nth term = a1(r n-1) • r = common ratio (what is being multiplied) Geometric Sequence Find the 7th term: N = 2(3n-1)

20. When to use: Equation: Geometric nth term rules

21. When to use: Use when the pattern increases or decreases through multiplying or dividing. Equation: ar(n-1) A = 1st term R = the ratio between each number N = term you are finding Geometric nth term rules

22. Find the Nth term rule for each sequence 1) 2) 3)

23. 1. 1, 4, 16, 64, … Find the 12th term 2. 5, 15, 45, 135, … Find the 7th term 3. -4, -8, -16, -32 … find the 9th term Find the nth term and then the missing numbers.

24. Assessment Prompt When do you use the arithmetic rule? Use the arithmetic rule when the sequence is changing by addition or subtraction. When do you use the geometric rule? Use the geometric rule when the sequence is changing by multiplication or division.

25. n 1 2 3 4 5 y -1 -4 -16 -64 In the sequence -1, -4, -16, -64, ,…, each number is multiplied by 4. Additional Example 1A: Identifying Patterns in a Sequence Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. -256 -64 ● 4 = -256. Multiply the fourth number by 4. The sequence is geometric. When n = 5, y = -256.

26. n 1 2 3 4 5 y 51 46 41 36 In the sequence 51, 46, 41, 36, ,…, -5 is added each time. Additional Example 1B: Identifying Patterns in a Sequence Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. 31 36 + (-5) = 31. Add -5 to the fourth number. The sequence is arithmetic. When n = 5, y = 31.

27. n 1 2 3 4 5 y 12 16 20 24 In the sequence 12, 16, 20, 24, ,…, 4 is added each time. Check It Out: Example 1A Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. 28 24 + 4 = 28. Add 4 to the fourth number. The sequence is arithmetic. When n = 5, y = 28.

28. n 1 2 3 4 5 y -1 -3 -9 -27 In the sequence -1, -3, -9, -27, ,…, each number is multiplied by 3. Check It Out: Example 1B Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. -81 -27 ● 3 = -81. Multiply the fourth number by 3. The sequence is geometric. When n = 5, y = -81.

29. n Rule y 1 3 2 6 3 9 4 12 Additional Example 2A: Identifying Functions in Sequences Write a function that describes the sequence. 3, 6, 9, 12,… Make a function table. 1 • 3 2 • 3 Multiply n by 3. 3 • 3 4 • 3 The function y = 3n describes this sequence.

30. n Rule y 1 4 2 7 3 10 4 13 Additional Example 2B: Identifying Functions in Sequences Write a function that describes the sequence. 4, 7, 10, 13,… Make a function table. 3(1) + 1 3(2) + 1 Multiply n by 3 and add 1. 3(3) + 1 3(4) + 1 The function y = 3n + 1describes this sequence.

31. n Rule y 1 5 2 6 3 7 4 8 Check It Out: Example 2A Write a function that describes the sequence. 5, 6, 7, 8,… Make a function table. 1 + 4 2 + 4 Add 4 to n. 3 + 4 4 + 4 The function y = 4 + n describes this sequence.

32. Additional Example 3: Using Functions to Extend Sequences Holli keeps a list showing her cumulative earnings for walking her neighbor’s dog. She recorded $1.25 the first time she walked the dog, $2.50 the second time, $3.75 the third time, and $5.00 the fourth time. Write a function that describes the sequence, and then use the function to predict her earnings after 9 walks. Write the number of walks she recorded; 1.25, 2.50, 3.75, 5.00. Make a function table.

33. n Rule y 1 1.25 2 2.50 3 3.75 4 5.00 Additional Example 3 Continued 1 • 1.25 Multiply n by 1.25. 2 • 1.25 3 • 1.25 4 • 1.25 Write the function. y = 1.25n 9 walks correspond to n = 9. When n = 9, y = 1.25 • 9 = 11.25. Holli would earn $11.25 after 9 walks.

34. Lesson Quiz: Part I Tell whether each sequence of y-values is arithmetic or geometric. Write a function that describes each sequence, and then find y when n = 5. 1.6, 12, 24, 48,… 2.–3, –2, –1, 0,… 3.24, 21, 18, 15,… geometric; an= 6(2n-1); 96 arithmetic;an= n – 4; 1 arithmetic; an= 27 – 3n; 12

35. Lesson Quiz for Student Response Systems 1. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5. 4, 8, 12, 16, … A. arithmetic; y = 4n; 20 B. geometric; y = 2n; 10 C. arithmetic; y = 4 + n; 9 D. arithmetic; y = 2n + 2; 12

36. Lesson Quiz for Student Response Systems 2. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5. –5, –4, –3, –2, … A. arithmetic; y = n + 6; 11 B. geometric; y = 2n – 6; 4 C. arithmetic; y = n – 6; –1 D. geometric; y = n; 5

37. Lesson Quiz for Student Response Systems 3. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5. 16, 12, 8, 4, … A. arithmetic; y = 20 – 2n; 10 B. geometric; y = 4n; 20 C. arithmetic; y = 20 – 4n; 0 D. geometric; y = 30 – 6n; 0

38. 3n – 2 • 5n • -2n + 1 • 7n Write the number sequence

39. Summarizer: