Arithmetic and Geometric Sequences

Demonstrate with your Smarties: 2, 5, 8, 11, 14, 17

1. How many groups of Smarties do you have?2. What did you add from one number to get the

next number?3. How many times did you add that number?4. Suppose you had continued the pattern for a total

of 24 times, how many times would you have added that number?

5. Suppose you had continue the pattern ‘n’ times, how many times would you have added that number?

6

3

5

23

n-1

If we had started with 7 smarties, what would the equation be?An = 2 + (n – 1)3An = 7 + (n – 1)3

If we had started with a1 smarties, what would the equation be?An = a1 + (n – 1)3

If we added 5 smarties each time, what would the equation be?

An = a1+ (n – 1)5 An = a1 + (n – 1)d

If we added d smarties each time, what would the equation be?

3 3 3 3 3

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Demonstrate with your Smarties: 2,5,8,11,14,17

1. Add the total number of Smarties2. Combine the 2 and 17; Add them3. Combine the 5 and 14; Add them 4. How many groups do you have?5. How many “pairs of 19” could you make?6. Multiply the number of pairs by ‘19’7. What if you had 12 numbers, how many

pairs could you make? 6. What if you had 5 numbers, how many

pairs could you make?7. What if you had n numbers, how many

pairs could you make?8. What is the sum of the 1st 100 numbers?

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57

19

6

3

57

6

2.5

n/2

5050

Demonstrate with your Smarties: 1, 2, 4, 8, 16, 32

1. What are you multiplying each number by to get the next number?

2. How many groups do you have?3. How many times did you multiply by 2 to get

the last number?4. Explain how the last number (32) can be

written as An=1*26-1

5. How many times would you multiply by 2 if there are 8 numbers?

6. How many times would you multiply by 2 if there are n numbers?

7. Rewrite #4 with using ‘n’s’

2

6

5

7

n-1

An=1*2n-1

8. What would the equation be if we started with 3?9. What is new equation if you started with a1?

An=3*2n-1

An=a1*2n-1

10. What is new equation if we multiplied by r?An=a1*rn-1

2 2 2 2 2

Demonstrate with your Smarties: 1,2,4,8,16,321. How many total Smarties do you have?2. How many groups do you have?3. What do you multiply by?4. Evaluate Sn= 1* 26 - 1 2 - 15. 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

8. 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

26

Sn =1* 25 - 1 2 – 1 = 31Sn = 1* 2n - 1 2 - 1

63

Sn = 1* 43 - 1 4 – 1 = 21Sn = 1* rn - 1 r - 1

Sn =a1 * rn - 1 r - 1

42

2 Sn = 2* 43 - 1 = 42 4 - 1

An = a1 + (n – 1)dFinding the nth number of arithmetic sequenceFinding 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 – 1Or Sn= a1 * 1- rn

1 – r

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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}