1. Lake Zurich High School
Geometric
Sequences & Series
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: October 11, 2005

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

3. nth term of geometric sequence
an = a1·r(n-1)
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4. 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

5. 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

6. 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

7. 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

8. 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

9. Sum it up
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10.
a1 = 1
r = 3
n = 6
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11. –
a1 = 4
r = -2
n = 7
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12. Alternative Sum Formula
We know that:
Multiply by r:
Simplify:
Substitute:
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13. Find the sum of the geometric Series
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14. Evaluate = 2 + 4 + 8+…+1024 a1 = 2 r = 2 n = 10 an = 1024
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15. Evaluate = 3 + 6 + 12 +…+ 384 a1 = 3 r = 2 n = 8 an = 384
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16. an = a1·r(n-1) Review -- Geometric Sum of n terms nth term
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17. Geometric
Infinite Series
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18. The Magic Flea (magnified for easier viewing)
There is no flea like a Magic Flea
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19. The Magic Flea (magnified for easier viewing)
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20. Sum it up -- Infinity
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21. Remember --The Magic Flea
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22. Jeff Bivin -- LZHS

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. Find the sum of the series
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32. Fractions - Decimals
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33. Let’s try again + +
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34. One more
subtract
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35. OK now a series
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36. .9 = 1
.9 = 1
That’s All Folks
Jeff Bivin -- LZHS