Download presentation

Presentation is loading. Please wait.

Published byKirk Cowdrey Modified over 2 years ago

1
An Interactive Tutorial by S. Mahaffey (Osborne High School)

2
Sequences are functions. They are made up of terms that go in a specific order. The position of each term in the sequence is called n. For the 5 th term, n = 5. For the 8 th term, n = 8. Each term is called a n ( pronounced “a sub n” ).

3
Consider the sequence 2, 4, 6, 8, … The first term is called a 1, “a sub 1”. For this example, a 1 = 2. The second term is 4, so a 2 = 4. The third term is 6, so a 3 = 6. The fourth term is 8, so a 4 = 8. The nth term is called a n.

4
Consider the sequence 15, 20, 25, 30, 35, … What is a 1 ? 15 20 25 30 35 What is a 3 ? 15 20 25 30 35 What is a 5 ? 15 20 25 30 35 click

5
The domain of a sequence is it’s n values. Since n is just the position of the term, then the domain is the whole numbers 1, 2, 3, 4, etc. The range of a sequence is it’s a n values, or terms.

6
Which is the correct domain and range for the sequence 3, 10, 17, 24, … D:{1,2, 3, 4,…} R:{3, 10, 17, 24,…} D:{3, 10, 17, 24,…} R:{1, 2, 3, 4, …} Click here

7
One more example… Which is the correct domain and range for the sequence 100, 75, 50, 25, 0, -25, … D:{100, 75, 50,,25, 0, -25, …} R:{1, 2, 3, 4, 5, 6, …} D:{1, 2, 3, 4, 5, 6, …} R:{100, 75, 50,,25, 0, -25, …} Click here

8
Sometimes the terms of the sequence are not listed. Instead, you are given an equation. If this is the case, you can find the first term by letting n=1, and the second term by letting n=2, and so on.

9
Find the domain and range of the sequence a n = 3( n -1) + 2. First, let’s write out the first 4 terms: a 1 = 3(1-1) + 2 = 3(0) + 2 = 2 a 2 = 3(2-1) + 2 = 3(1) + 2 = 5 a 3 = 3(3-1) + 2 = 3(2) + 2 = 8 a 4 = 3(4-1) + 2 = 3(3) + 2 = 11 So, the domain is {1, 2, 3, 4, …} and the range is {2, 5, 8, 11, …}

10
Find the domain and range for a n = 2(n-1) + 10 D: {1, 2, 3, 4, …} R: {2, 4, 6, 8, …} D: {1, 2, 3, 4, …} R: {10, 20, 30, 40, …} D: {1, 2, 3, 4, …} R: {10, 12, 14, 16, …} Click here

11
Find the domain and range for a n = -4(n-1) + 1 D: {1, 2, 3, 4, …} R: {-2, -4, -6, -8, …} D: {1, 2, 3, 4, …} R: {1, -3, -7, -11, …} D: {1, 2, 3, 4, …} R: {-4, -8, -16, -32, …} Click here

12
Find the domain and range for a n = 5 ∙ 2 (n-1) D: {1, 2, 3, 4, …} R: {5, 10, 20, 40, …} D: {1, 2, 3, 4, …} R: {5, 10, 15, 20, …} D: {1, 2, 3, 4, …} R: {1, 10, 100, 1000, …} Click here

13
In the last 3 examples, the sequences were given as formulas and you had to calculate the terms. Now, we’re going to focus on how to write the formulas when you are given the terms.

14
There are two types of equations we’ll be looking at in this tutorial. The first type of formula is called a recursive formula. The second type of formula is called a closed formula.

15
Recursive formulas are the easiest to write. Each term is defined by the term that comes before it. (The previous term). Recursive formulas have 2 parts: define the first term define the next term by its relationship to the previous term. For example, in the sequence 90, 95, 100, 105, … the first term is 90 and the next term is five plus the previous term. In the sequence 14, 12, 10, 8, … the first term is 14 and the next term is two minus the previous term. In the sequence 1, 2, 4, 8, 16, … the first term is 1 and the next term is 2 times the previous term.

16
English Math Symbol The next termanan The previous terma n-1 Recursive formulas are written almost exactly like you’d say it in English. But instead of writing out phrases like “the next term”, you’ll use these math symbols instead:

17
Write the recursive equation for the series 9, 11, 13, 15, … Identify the pattern: The first term is 9 The next term is equal to the previous term plus 2. Write the equation: a 1 = 9 a n = a n-1 + 2

18
Write the recursive equation for the series 25, 30, 35, 40, 45… Identify the pattern: The first term is 25. The next term is equal to the previous term plus 5. Write the equation: a 1 = 25 a n = a n-1 + 5

19
Write the recursive equation for the series 10, 9, 8, 7, 6… Identify the pattern: The first term is 10. The next term is equal to the previous term minus 1. Write the equation: a 1 = 10 a n = a n-1 - 1

20
Write the recursive equation for the series 3, 6, 12, 24, 48… Identify the pattern: The first term is 3. The next term is equal to the previous term times 2. Write the equation: a 1 = 3 a n = a n-1 ∙ 2

21
You try: Choose the correct recursive formula for the sequence 3, 7, 11, 15, 19, … a 1 = 3a 1 = 3 a n = a n+1 + 4a n = a n-1 + 4 a 1 = 4a 1 = 4 a n = a n+1 + 4a n = a n-1 + 4

22
You try: Choose the correct recursive formula for the sequence 19, 15, 11, 7, 3, … a 1 = 19a 1 = 4 a n = a n-1 + 4a n = a n-1 + 19 a 1 = 19a 1 = -4 a n = a n-1 - 4a n = a n-1 - 19

23
You try: Choose the correct recursive formula for the sequence 7, 14, 28, 56, 112, … a 1 = 7a 1 = 7 a n = a n-1 ∙ 2 a n = a n-1 + 7 a 1 = 2a 1 = 2 a n = a n-1 + 7a n = a n-1 ∙ 7

24
You try: Choose the correct recursive formula for the sequence 1, 8, 64, 512, … a 1 = 8a 1 = 1 a n = a n-1 ∙ 8 a n = a n-1 + 8 2 a 1 = 8a 1 = 1 a n = a n-1 ∙ 1 a n = a n-1 ∙ 8

25
Recursive formulas are easy to write, but if you want to find the 50 th term, a 50, you have to know the first 49 terms. This can be time consuming. With closed formulas, you can easily find the 50 th term (n = 50). The formula uses the variable n instead of a n-1 (which is the value of the previous term). So, if you want the 10 th term, then n=10. If you want the 20 th term, then n=20…

26
When you add or subtract to get to the next term, then you have an arithmetic sequence: To write a closed formula, start with the recursive formula: a 1 = a n = a n-1 + Then rearrange the equation to get the closed formulas: a n = (n-1) +

27
Arithmetic: Recursive a 1 = 3 a n = a n-1 + 4 Closed a n = 4 (n-1) + 3

28
Find the 101 st term of the sequence 2, 6, 10, 14, … First write the recursive formula (The first term is 2 and each additional term is the previous term plus 4) a 1 = 2 a n = a n-1 + 4 Next, write the closed formula: a n = 4(n-1) + 2 Finally, find the 101 st term: a 101 = 4(101-1) + 2 = 4(100) + 2 = 402

29
Find the 51 st term of the sequence 12, 15, 18, 21, … First write the recursive formula (The first term is 12 and each additional term is the previous term plus 3) a 1 = 12 a n = a n-1 + 3 Next, write the closed formula: a n = 3(n-1) + 12 Finally, find the 51 st term: a 101 = 3(51-1) + 12 = 3(50) + 12 = 162

30
Find the 81 st term of the sequence 100, 90, 80, 70, … First write the recursive formula (The first term is 100 and each additional term is the previous term minus 10) a 1 = 100 a n = a n-1 - 10 Next, write the closed formula: a n = -10(n-1) + 100 Finally, find the 81 st term: a 101 = -10(81-1) + 100 = -10(80) + 100 = -700

31
Write the closed formula for the sequence: a 1 = 3 a n = a n-1 + 8 a n = 8(a n-1 ) + 3a n = 3(a n-1 ) + 8 a n = 8(n-1) + 3a n = 3(n-1) + 8

32
Write the closed formula for the sequence: a 1 = 15 a n = a n-1 - 2 a n = 2(n-1) + 15a n = -2(n-1) + 15 a n = -2(n-1) - 15a n = 2(n-1) - 15

33
Write the closed formula for the sequence: 15, 20, 25, 30, … a n = 15(n-1) - 5a n = 15(n-1) + 5 a n = 5(n-1) - 15a n = 5(n-1) + 15

34
Find the 101 st term of the sequence: 8, 11, 14, 17, 20, … a 101 = 803a 101 = 308 a 51 = 158a 100 = 305

35
Find the 51 st term of the sequence: 81, 79, 77, 75, 73, … a 101 = -19a 101 = -119 a 51 = -19a 51 = -119

36
When you multiply or divide to get to the next term, then you have a geometric sequence: To write a closed formula, start with the recursive formula: a 1 = a n = a n-1 ∙ Then rearrange the equation to get the closed formulas: a n = ∙ (n-1)

37
Geometric: Recursive a 1 = 3 a n = a n-1 ∙ 4 Closed a n = 3 ∙ 4 (n-1)

38
Find the 9 th term of the sequence 1, 2, 4, 8, 16, … First write the recursive formula (The first term is 1 and each additional term is the previous term times 2) a 1 = 1 a n = a n-1 ∙ 2 Next, write the closed formula: a n = 1 ∙ 2 (n-1) Finally, find the 9 th term: a 9 = 1 ∙2 8 = 1∙256 = 256

39
Find the 10 th term of the sequence 2, 6, 18, 54, … First write the recursive formula (The first term is 2 and each additional term is the previous term times 3) a 1 = 2 a n = a n-1 ∙ 3 Next, write the closed formula: a n = 2 ∙ 3 (n-1) Finally, find the 10 th term: a 10 = 2 ∙ 3 9 = 2 ∙19,683 = 39,366

40
Write the closed formula for the sequence: a 1 = 3 a n = a n-1 ∙ 8 a n = 3 ∙ 8 (an-1) a n = 8 ∙ 3 (n-1) a n = 3 ∙ 8 (n-1) a n = 24 (n-1)

41
Write the closed formula for the sequence: a 1 = 15 a n = a n-1 ∙ 2 a n = 2 ∙15a n = 15 ∙ 2 (n-1) a n = 30 (n-1) a n = 2(n-1) ∙ 15

42
Write the closed formula for the sequence: 5, 15, 45, 135, … a n = 3 ∙5(n-1) a n = 3 ∙5 (n-1) a n = 5 ∙3(n-1) a n = 5 ∙ 3 (n-1)

43
Find the 8 th term of the sequence: 1, 3, 9, 27, 81, … a 8 = 2,187a 7 = 729 a 8 = 6,561a 7 = 2,187

44
Find the 10 th term of the sequence: 2, 10, 50, 250, 1250, … a 10 = 1,953,125a 9 = 1,953,125 a 10 = 3,906,250a 9 = 781,250

46
Oops, wrong answer… Try Again… Remember a 4 will be the fourth term in the sequence, a 2 will be the second term, and a n will be the n th term…

48
Oops, wrong answer… Try Again… Remember, the domain is the set of whole numbers… The range is the terms of the sequence…

49
Oops, wrong answer… Try Again… Remember, a 1 is the first term. plus something a n = a n-1 minus something times something divided by something

50
Oops, wrong answer… For arithmetic sequences, where you add or subtract to get to next term: Recursive formula:a 1 = a n = a n-1 + Closed formula: a n = (n-1) +

51
Oops, wrong answer… For geometric sequences, where you multiply or divide to get to next term: Recursive formula:a 1 = a n = a n-1 ∙ Closed formula: a n = ∙ (n-1)

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google