1. GEOMETRIC
SERIES

2. There is a legend that a Persian invented chess to give interest to the life of the king who was bored. For his reward, this Perian asked for a quantity of grain, according to the following rules.

3. 1 grain was to be placed on the 1st square of the chess board,
2 on the next,
4 on the 3rd
and so on, doubling the number each square.
How many must be placed on the 64th square?

4. Geometric Sequence
We have a sequence:
Each term is twice the previous term, so by the 64th term we have multiplied by 2 sixty-three times
We have
approximately or 9 followed by 18 zeros!

5. The sequence
is an example of a
Geometric sequence
A sequence is geometric if
where r is a constant called the common ratio
In the above sequence, r = 2

6. A geometric sequence or geometric progression (G.P.) is of the form
The nth term of an G.P. is

7. Exercises
1. Use the formula for the nth term to find the term indicated of the following geometric sequences
(a)
Ans:
(b)
Ans:
(c)
Ans:

8. The formula will be proved next but you don’t need to learn the proof.
Summing terms of a G.P.
e.g.1 Evaluate
Writing out the terms helps us to recognize the G.P.
Although with a calculator we can see that the sum is 186, we need a formula that can be used for any G.P.
The formula will be proved next but you don’t need to learn the proof.

9. Move the lower row 1 place to the right (Combining like terms)
Summing terms of a G.P.
With 5 terms of the general G.P., we have
Multiply by r:
Subtracting the expressions gives
Move the lower row 1 place to the right (Combining like terms)

10. Summing terms of a G.P.
With 5 terms of the general G.P., we have
Multiply by r:
Subtracting the expressions gives
and subtract

11. Summing terms of a G.P.
With 5 terms of the general G.P., we have
Multiply by r:
Subtracting the expressions gives

12. Summing terms of a G.P.
So,
Take out the common factors
and divide by ( 1 – r )
Similarly, for n terms we get

13. Sum of a FINITE Geometric Series.
Number of terms
First term
Rate “Common Ratio”
Sum of “n” terms
What is “missing”?
There is no last term

14. Summing terms of a G.P.
gives a negative denominator if r > 1
The formula
Instead, we can use
To get this version of the formula, we’ve multiplied the 1st form by

15. Summing terms of a G.P.
For our series
Using

16. Summing terms of a G.P.
e.g. 2
Find the sum of the first 20 terms of the geometric series, leaving your answer in index form
Solution:
We’ll simplify this answer without using a calculator

17. Summing terms of a G.P.
There are 20 minus signs here and 1 more outside the bracket!

18. Evaluate the geometric series: 4 + 2 + 1 + ½ + …
Summing terms of a G.P.
Evaluate the geometric series: ½ + …
For the first 10 terms

19. Evaluate the geometric series: 4 + 2 + 1 + ½ + …
Summing terms of a G.P.
Evaluate the geometric series: ½ + …
For the first 10 terms

20. Evaluate the geometric series: 4 + 2 + 1 + ½ + …
Summing terms of a G.P.
Evaluate the geometric series: ½ + …
For the first 10 terms

21. Evaluate the geometric series: 4 + 2 + 1 + ½ + …
Summing terms of a G.P.
Evaluate the geometric series: ½ + …
For the first 10 terms

22. Evaluate the geometric series: 4 + 2 + 1 + ½ + …
Summing terms of a G.P.
Evaluate the geometric series: ½ + …
For the first 10 terms

23. Evaluate the geometric series: 4 + 2 + 1 + ½ + …
Summing terms of a G.P.
Evaluate the geometric series: ½ + …
For the first 10 terms

24. Evaluate the geometric series: 4 + 2 + 1 + ½ + …
Summing terms of a G.P.
Evaluate the geometric series: ½ + …
For the first 10 terms

25. Evaluate the geometric series: 4 + 2 + 1 + ½ + …
Summing terms of a G.P.
Evaluate the geometric series: ½ + …
For the first 10 terms

26. SUMMARY
A geometric sequence or geometric progression (G.P.) is of the form
The nth term of an G.P. is
The sum of n terms is or

28. GEOMETRIC
SERIES

29. Suppose we have a 2 metre length of string . . .
. . . which we cut in half
We leave one half alone and cut the 2nd in half again
. . . and again cut the last piece in half

30. Continuing to cut the end piece in half, we would have in total
In theory, we could continue for ever, but the total length would still be 2 metres, so
This is an example of an infinite series.

31. The series
is a G.P. with the common ratio
Even though there are an infinite number of terms, this series converges to 2.
Number of terms, n
Sum

32. The series
is a G.P. with the common ratio
Even though there are an infinite number of terms, this series converges to 2.

33. Where does each term approach? What happens to each term in the series?

34. This is NOT hitting the x-axis
Where does each term approach? What happens to each term in the series?
The series
This is NOT hitting the x-axis

35. The series
The General Term is:

36. An infinite series is an expression of the form:

37. Summing terms of a G.P.

38. Converging or Diverging?

39. Converging or Diverging?

40. Converging or Diverging?
To where?

41. y= 1.5

42. Arithmetic and Geometric Series

43. Definition of a Geometric Series

44. Why do 1, 3, 4 and 6 Diverge?

45. We will find a formula for the sum of an infinite number of terms of a G.P. This is called “the sum to infinity”,
e.g. For the G.P.
we know that the sum of n terms is given by
As n varies, the only part that changes is
This term gets smaller as n gets larger.

46. If |r| < 1, then the infinite geometric series
The sum of the terms of an infinite geometric sequence is called a geometric series.
If |r| < 1, then the infinite geometric series
a1 + a1r + a1r2 + a1r a1rn
has the sum

47. Example: Find the sum of
The sum of the series is

48. The sum of a finite geometric sequence is given by
= ?
n = 8
a1 = 5

49. Convergent and Divergent Series
If the infinite series has a sum the series is convergent.
If the series is not convergent, it is divergent.

50. Ways To Determine Convergence/Divergence
1. Arithmetic – since no sum exists, it diverges
2. Geometric:
If |r| > 1, diverges
If |r| < 1, converges since the sum exists

51. Example
Determine whether each arithmetic or geometric series is convergent or divergent.
1/8 + 3/20 + 9/ /
r=6/5  |r|>1  divergent
Arithmetic series  divergent
/5 + 13/
r=1/5  |r|<1  convergent

52. Other Series
When a series is neither arithmetic or geometric, it is more difficult to determine whether the series is convergent or divergent.

53. Convergent Geometric Series
Where a is the first term and r is the constant ratio.

54. As n approaches infinity, approaches zero.
We write:
So, for ,
For the series

55. However, if, for example r = 2,
As n increases, also increases. In fact,
The geometric series with diverges
There is no sum to infinity

56. Convergence
If r is any value greater than 1, the series diverges.
Also, if r < -1, ( e.g. r = -2 ),
So, again the series diverges.
If r = 1, all the terms are the same.
If r = -1, the terms have the same magnitude but they alternate in sign. e.g. 2, -2, 2, -2, . . .
A Geometric Series converges only if the common ratio r lies between -1 and 1.
for
This can also be written as

57. e.g. 1. For the following geometric series, write down the value of the common ratio, r, and decide if the series converges. If so, find the sum to infinity.
Solution:
so r does satisfy -1 < r < 1
The series converges to

58. SUMMARY
A geometric sequence or geometric progression (G.P.) is of the form
The nth term of an G.P. is
The sum of n terms is or
The sum to infinity is or

59. Exercises
1. For the following geometric series, write down the value of the common ratio, r, and decide if the series converges. If so, find the sum to infinity.
Ans: (a) so the series diverges.
(b) so the series converges.