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Arithmetic and Geometric Sequences. Demonstrate with your Smarties: 2 , 5, 8, 11, 14, 17. 3. 3. 3. 3. 3. 6. How many groups of Smarties do you have? What did you add from one number to get the next number? How many times did you add that number?
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Demonstrate with your Smarties: 2, 5, 8, 11, 14, 17 3 3 3 3 3 6 • How many groups of Smarties do you have? • What did you add from one number to get the next number? • How many times did you add that number? • Suppose you had continued the pattern for a total of 24 times, how many times would you have added that number? • Suppose you had continue the pattern ‘n’ times, how many times would you have added that number? 3 5 23 n-1 An = a1 + (n – 1)d An = 2 + (n – 1)3 An = 7 + (n – 1)3 An = a1 + (n – 1)3 An = a1+ (n – 1)5 If we had started with 7 smarties, what would the equation be? If we had started with a1 smarties, what would the equation be? If we added 5 smarties each time, what would the equation be? If we added d smarties each time, what would the equation be?
Demonstrate with your Smarties: 2,5,8,11,14,17 57 • Add the total number of Smarties • Combine the 2 and 17; Add them • Combine the 5 and 14; Add them • How many groups do you have? • How many “pairs of 19” could you make? • Multiply the number of pairs by ‘19’ • What if you had 12 numbers, how many pairs could you make? • What if you had 5 numbers, how many pairs could you make? • What if you had n numbers, how many pairs could you make? • What is the sum of the 1st 100 numbers? 19 6 3 57 6 2.5 n/2 5050 :http://www.rare-earth-magnets.com/t-johann-carl-friedrich-gauss.aspx
Demonstrate with your Smarties: 1, 2, 4, 8, 16, 32 2 2 2 2 2 2 • What are you multiplying each number by to get the next number? • How many groups do you have? • How many times did you multiply by 2 to get the last number? • Explain how the last number (32) can be written as An=1*26-1 • How many times would you multiply by 2 if there are 8 numbers? • How many times would you multiply by 2 if there are n numbers? • Rewrite #4 with using ‘n’s’ 6 5 7 n-1 An=1*2n-1 An=3*2n-1 8. What would the equation be if we started with 3? 9. What is new equation if you started with a1? An=a1*2n-1 10. What is new equation if we multiplied by r? An=a1*rn-1
Demonstrate with your Smarties: 1,2,4,8,16,32 • How many total Smarties do you have? • How many groups do you have? • What do you multiply by? • Evaluate Sn=1* 26- 1 • 2 - 1 • 5. Remove the 32 smarties. Write & evaluate the new equation and verify with Smarties. • 6. If there were ‘n’ groups of numbers, what would the new equation be? • 7. Suppose we were to multiply by 4, how would the equation in #6 be different? Verify by using 1,4,16 only • If we multiplied by “r” then what would the equation be? • 9. If we started the sequence in #7 with 2 instead of 1? • a. How many total Smarties would there be? • b. What do you multiply the equation in #7 by? • 10. If you started with a1 in equation 8, what would the • new equation be? 63 6 2 63 Sn =1* 25- 1 2 – 1 = 31 Sn = 1* 2n- 1 2 - 1 Sn = 1* 43- 1 4 – 1 = 21 Sn = 1* rn- 1 r - 1 42 Sn = 2* 43- 1 = 42 4 - 1 2 Sn =a1*rn- 1 r - 1
An = a1 + (n – 1)d Finding the nth number of arithmetic sequence Finding the sum of n terms of an arithmetic sequence Sn = (n/2)(a1 + an) Finding the nth number of geometric sequence An=a1*rn-1 Finding the sum of n terms of a geometric sequence Sn =a1*rn- 1 r – 1 Or Sn= a1*1- rn 1 – r I want 1 sub right now, 1 sub Roger
1. 5, -3, -1, -9, -7, -15…. 2. 2, 11, 27, 51, 83…. 3. 6, 37, 50, 1, 2, 5, 26…. 4. {8, 4, 5, 9, 1, 7, 6, 3, 2, 0}
