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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 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. 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. 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 …. 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 When to use: Equation: Arithmetic nth term rules Equation: When to use: Use when the pattern is increasing or decreasing by addition or subtraction. 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 You can use a variable such as n, to represent a number’s position in a sequence. n (position in the sequence) 1 2 3 4 y (value of term) 2 4 6 8 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. Step 3 nth term formula is a + d(n-1) Find the Nth term rule for each sequence N: 1 2 3 4 5 Term: 5 8 11 14 17 N: 1 2 3 4 5 Term: 5 6 7 8 9 N: 1 2 3 4 5 Term: 2 7 12 17 22 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. 1. 6, 12, 18, 24,… 2. 24, 21, 18, 15,… 3. -28, -24, -20… 4. 45, 42, 39… Find the nth term expression 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. How is finding the nth term different than finding a function rule? Assessment Prompt From yesterday, What does a, d, and n stand for? Activator In a geometric sequence, each term is multiplied by the same amount to get the next term in the sequence. There are two types of patterns A sequence where the numbers increase by multiplication. Formula: nth term = a1(r n-1) r = common ratio (what is being multiplied) n (position in the sequence) 1 2 y (value of term) 2 6 Geometric Sequence th n-1 Find the 7 term: N = 2(3 ) 3 4 18 54 When to use: Equation: Geometric nth term rules When to use: Use when the pattern increases or decreases through multiplying or dividing. Geometric nth term rules Equation: ar(n-1) A = 1st term R = the ratio between each number N = term you are finding 1) 2) 3) Find the Nth term rule for each sequence N: 1 2 3 4 5 Term: 5 10 20 40 80 N: 1 2 3 4 5 Term: 3 6 12 24 48 N: 1 2 3 4 5 Term: 4 8 16 32 64 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. 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. Assessment Prompt 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. n 1 2 3 y -1 -4 -16 4 5 -64 -256 In the sequence -1, -4, -16, -64, number is multiplied by 4. ,…, each -64 ● 4 = -256. Multiply the fourth number by 4. The sequence is geometric. When n = 5, y = -256. 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. n 1 2 3 4 5 y 51 46 41 36 31 In the sequence 51, 46, 41, 36, each time. 36 + (-5) = 31. ,…, -5 is added Add -5 to the fourth number. The sequence is arithmetic. When n = 5, y = 31. Check It Out: Example 1A Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. n 1 2 3 4 5 y 12 16 20 24 28 In the sequence 12, 16, 20, 24, each time. 24 + 4 = 28. ,…, 4 is added Add 4 to the fourth number. The sequence is arithmetic. When n = 5, y = 28. Check It Out: Example 1B Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. n 1 2 3 y -1 -3 -9 4 5 -27 -81 In the sequence -1, -3, -9, -27, number is multiplied by 3. -27 ● 3 = -81. ,…, each Multiply the fourth number by 3. The sequence is geometric. When n = 5, y = -81. Additional Example 2A: Identifying Functions in Sequences Write a function that describes the sequence. 3, 6, 9, 12,… Make a function table. n Rule y 1 1•3 3 2 2•3 6 3 3•3 9 4 4•3 12 Multiply n by 3. The function y = 3n describes this sequence. Additional Example 2B: Identifying Functions in Sequences Write a function that describes the sequence. 4, 7, 10, 13,… Make a function table. n Rule y 1 3(1) + 1 4 2 3(2) + 1 7 3 3(3) + 1 10 4 3(4) + 1 13 Multiply n by 3 and add 1. The function y = 3n + 1describes this sequence. Check It Out: Example 2A Write a function that describes the sequence. 5, 6, 7, 8,… Make a function table. n Rule y 1 1+4 5 2 2+4 6 3 3+4 7 4 4+4 8 Add 4 to n. The function y = 4 + n describes this sequence. 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. Additional Example 3 Continued n Rule y 1 1 • 1.25 1.25 2 2 • 1.25 2.50 3 3 • 1.25 3.75 4 4 • 1.25 5.00 y = 1.25n Multiply n by 1.25. Write the function. 9 walks correspond to n = 9. When n = 9, y = 1.25 • 9 = 11.25. Holli would earn $11.25 after 9 walks. 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,… geometric; an = 6(2n-1); 96 2. –3, –2, –1, 0,… arithmetic; an= n – 4; 1 3. 24, 21, 18, 15,… arithmetic; an = 27 – 3n; 12 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 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 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 1. 3n – 2 2. 5n 3. -2n + 1 4. 7n Write the number sequence Review: Give an example of each: natural #, whole #, integer, rational, and irrational Identify the sequence 7,11,15,19 Find the 32nd term of the sequence to the left Identify the sequence 500, 250,125,62.5 Review: Chris bought a jet ski for $7500. He Find the 8th term of the sold it 4 years later for sequence to the left $2300. Find the % of change Identify the sequence 6,-12,24,-48 Find the 11th term of the sequence to the left Summarizer: Review: Identify as direct or inverse and the constant: 5 8 11 14 15 24 33 42