1. Lake Zurich High School
Arithmetic
Sequences & Series
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: April 28, 2006

2. Arithmetic Sequences 5, 8, 11, 14, 17, 20, … 3n+2, …
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3. nth term of arithmetic sequence
an = a1 + d(n – 1)
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4. Find the nth term of an arithmetic sequence
First term is 8
Common difference is 3
an = a1 + d(n – 1)
an = 8 + 3(n – 1)
an = 8 + 3n – 3
an = 3n + 5
Jeff Bivin -- LZHS

5. an = a1 + d(n – 1) an = -6 + 7(n – 1) an = -6 + 7n – 7 an = 7n - 13
Finding the nth term
First term is -6
common difference is 7
an = a1 + d(n – 1)
an = (n – 1)
an = n – 7
an = 7n - 13
Jeff Bivin -- LZHS

6. an = a1 + d(n – 1) an = 23 + -4(n – 1) an = 23 - 4n + 4 an = -4n + 27
Finding the nth term
First term is 23
common difference is -4
an = a1 + d(n – 1)
an = (n – 1)
an = n + 4
an = -4n + 27
Jeff Bivin -- LZHS

7. an = a1 + d(n – 1) a100 = 5 + 6(100 – 1) a100 = 5 + 6(99)
Finding the 100th term
5, 11, 17, 23, 29, . . .
an = a1 + d(n – 1)
a100 = 5 + 6(100 – 1)
a100 = 5 + 6(99)
a100 =
a100 = 599
a1 = 5
d = 6
n = 100
Jeff Bivin -- LZHS

8. an = a1 + d(n – 1) a956 = 156 + -16(956 – 1) a956 = 156 - 16(955)
Finding the 956th term
a1 = 156
d = -16
n = 956
156, 140, 124, 108,
an = a1 + d(n – 1)
a956 = (956 – 1)
a956 = (955)
a956 =
a956 =
Jeff Bivin -- LZHS

9. Find the Sum of the integers from 1 to 100
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10. Summing it up Sn = a1 + (a1 + d) + (a1 + 2d) + …+ an
Sn = an + (an - d) + (an - 2d) + …+ a1
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11.
a1 = 1
an = 19
n = 7
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12.
a1 = 4
an = 24
n = 11
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13. Find the sum of the integers from 1 to 100
a1 = 1
an = 100
n = 100
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14. Find the sum of the multiples of 3 between 9 and 1344
Sn =
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15. Find the sum of the multiples of 7 between 25 and 989
Sn =
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16. Evaluate Sn = 16 + 19 + 22 + . . . + 82 a1 = 16 an = 82 d = 3 n = 23
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17. Evaluate Sn = -29 - 31 - 33 + . . . - 199 a1 = -29 an = -199 d = -2
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18. Find the sum of the multiples of 11 that are 4 digits in length
an = 9999
d = 11
Sn =
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19. Review -- Arithmetic
Sum of n terms
nth term
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20. Lake Zurich High School
Geometric
Sequences & Series
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: October 11, 2005

21. Geometric Sequences 1, 2, 4, 8, 16, 32, … 2n-1, …
81, 54, 36, 24, 16, … , . . .
Jeff Bivin -- LZHS

22. nth term of geometric sequence
an = a1·r(n-1)
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23. Find the nth term of the geometric sequence
First term is 2
Common ratio is 3
an = a1·r(n-1)
an = 2(3)(n-1)
Jeff Bivin -- LZHS

24. Find the nth term of a geometric sequence
First term is 128
Common ratio is (1/2)
an = a1·r(n-1)
Jeff Bivin -- LZHS

25. Find the nth term of the geometric sequence
First term is 64
Common ratio is (3/2)
an = a1·r(n-1)
Jeff Bivin -- LZHS

26. an = a1·r(n-1) an = 3·(2)10-1 an = 3·(2)9 an = 3·(512) an = 1536
Finding the 10th term
3, 6, 12, 24, 48, . . .
a1 = 3
r = 2
n = 10
an = a1·r(n-1)
an = 3·(2)10-1
an = 3·(2)9
an = 3·(512)
an = 1536
Jeff Bivin -- LZHS

27. an = a1·r(n-1) an = 2·(-5)8-1 an = 2·(-5)7 an = 2·(-78125)
Finding the 8th term
2, -10, 50, -250, 1250, . . .
a1 = 2
r = -5
n = 8
an = a1·r(n-1)
an = 2·(-5)8-1
an = 2·(-5)7
an = 2·(-78125)
an =
Jeff Bivin -- LZHS

28. Sum it up
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29.
a1 = 1
r = 3
n = 6
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30. –
a1 = 4
r = -2
n = 7
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31. Alternative Sum Formula
We know that:
Multiply by r:
Simplify:
Substitute:
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32. Find the sum of the geometric Series
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33. Evaluate = 2 + 4 + 8+…+1024 a1 = 2 r = 2 n = 10 an = 1024
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34. Evaluate = 3 + 6 + 12 +…+ 384 a1 = 3 r = 2 n = 8 an = 384
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35. an = a1·r(n-1) Review -- Geometric Sum of n terms nth term
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36. Geometric
Infinite Series
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37. The Magic Flea (magnified for easier viewing)
There is no flea like a Magic Flea
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38. The Magic Flea (magnified for easier viewing)
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39. Sum it up -- Infinity
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40. Remember --The Magic Flea
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41. Jeff Bivin -- LZHS

42. rebounds ½ of the distance from which it fell --
A Bouncing Ball
rebounds ½ of the distance from which it fell --
What is the total vertical distance that the ball traveled before coming to rest if it fell from the top of a 128 feet tall building?
128 ft
64 ft
32 ft
16 ft
8 ft
Jeff Bivin -- LZHS

43. A Bouncing Ball 128 ft 64 ft 32 ft 16 ft 8 ft Downward
= …
128 ft
64 ft
32 ft
16 ft
8 ft
Jeff Bivin -- LZHS

44. A Bouncing Ball 128 ft 64 ft 32 ft 16 ft 8 ft Upward
= …
128 ft
64 ft
32 ft
16 ft
8 ft
Jeff Bivin -- LZHS

45. A Bouncing Ball 128 ft 64 ft 32 ft 16 ft 8 ft
Downward = … = 256
Upward = … = 128
TOTAL = 384 ft.
128 ft
64 ft
32 ft
16 ft
8 ft
Jeff Bivin -- LZHS

46. rebounds 3/5 of the distance from which it fell --
A Bouncing Ball
rebounds 3/5 of the distance from which it fell --
What is the total vertical distance that the ball traveled before coming to rest if it fell from the top of a 625 feet tall building?
625 ft
375 ft
225 ft
135 ft
81 ft
Jeff Bivin -- LZHS

47. A Bouncing Ball 625 ft 375 ft 225 ft 135 ft 81 ft Downward
= …
625 ft
375 ft
225 ft
135 ft
81 ft
Jeff Bivin -- LZHS

48. A Bouncing Ball 625 ft 375 ft 225 ft 135 ft 81 ft Upward
= …
625 ft
375 ft
225 ft
135 ft
81 ft
Jeff Bivin -- LZHS

49. A Bouncing Ball 625 ft 375 ft 225 ft 135 ft 81 ft
Downward = … =
Upward = … = 937.5
TOTAL = 2500 ft.
625 ft
375 ft
225 ft
135 ft
81 ft
Jeff Bivin -- LZHS

50. Find the sum of the series
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51. Fractions - Decimals
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52. Let’s try again + +
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53. One more
subtract
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54. OK now a series
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55. .9 = 1
.9 = 1
That’s All Folks
Jeff Bivin -- LZHS