1. Sum Of A Geometric
Series

2. Sum Of A Geometric
Series
S n  a  ar  ar 2    ar n1

3. Sum Of A Geometric
Series
S n  a  ar  ar 2    ar n1
rSn  ar  ar 2  ar 3    ar n1  ar n

4. Sum Of A Geometric
Series
S n  a  ar  ar 2    ar n1
rSn  ar  ar 2  ar 3    ar n1  ar n
r 1Sn  ar n  a

5. Sum Of A Geometric
Series
S n  a  ar  ar 2    ar n1
rSn  ar  ar 2  ar 3    ar n1  ar n
r 1Sn  ar n  a
ar n  1
Sn 
r 1

6. Sum Of A Geometric
Series
S n  a  ar  ar 2    ar n1
rSn  ar  ar 2  ar 3    ar n1  ar n
r 1Sn  ar n  a
ar n  1
Sn  , if r  1
r 1

7. Sum Of A Geometric
Series
S n  a  ar  ar 2    ar n1
rSn  ar  ar 2  ar 3    ar n1  ar n
r 1Sn  ar n  a
ar n  1
Sn  , if r  1
r 1
OR
a1  r n 
Sn 
1 r

8. Sum Of A Geometric
Series
S n  a  ar  ar 2    ar n1
rSn  ar  ar 2  ar 3    ar n1  ar n
r 1Sn  ar n  a
ar n  1
Sn  , if r  1
r 1
OR
a1  r n  , if r  1
Sn 
1 r

9. Sum To Infinity (Limiting Sum)

10. Sum To Infinity (Limiting Sum)
NOTE : r  1

11. Sum To Infinity (Limiting Sum)
NOTE : r  1
If r  1, lim r n  0
n

12. Sum To Infinity (Limiting Sum)
NOTE : r  1
If r  1, lim r n  0
a1  r n 
n
lim S n  lim
n n 1  r

13. Sum To Infinity (Limiting Sum)
NOTE : r  1
If r  1, lim r n  0
a1  r n 
n
lim S n  lim
n n 1  r
a

1 r

14. Sum To Infinity (Limiting Sum)
NOTE : r  1
If r  1, lim r n  0
a1  r n 
n
lim S n  lim
n n 1  r
a

1 r
a
S  , if r  1
1 r

15. Sum To Infinity (Limiting Sum)
NOTE : r  1
If r  1, lim r n  0
a1  r n 
n
lim S n  lim
n n 1  r
a

1 r
a
S  , if r  1
1 r
e.g. i  Find the sum of the first 10 terms of 2  6  18  

16. Sum To Infinity (Limiting Sum)
NOTE : r  1
If r  1, lim r n  0
a1  r n 
n
lim S n  lim
n n 1  r
a

1 r
a
S  , if r  1
1 r
e.g. i  Find the sum of the first 10 terms of 2  6  18  
a  2, r  3 and n  10

17. Sum To Infinity (Limiting Sum)
NOTE : r  1
If r  1, lim r n  0
a1  r n 
n
lim S n  lim
n n 1  r
a

1 r
a
S  , if r  1
1 r
e.g. i  Find the sum of the first 10 terms of 2  6  18  
a  2, r  3 and n  10 ar n  1
Sn 
r 1

18. Sum To Infinity (Limiting Sum)
NOTE : r  1
If r  1, lim r n  0
a1  r n 
n
lim S n  lim
n n 1  r
a

1 r
a
S  , if r  1
1 r
e.g. i  Find the sum of the first 10 terms of 2  6  18  
a  2, r  3 and n  10 ar n  1
Sn 
r 1
2310  1
S10 
3 1

19. Sum To Infinity (Limiting Sum)
NOTE : r  1
If r  1, lim r n  0
a1  r n 
n
lim S n  lim
n n 1  r
a

1 r
a
S  , if r  1
1 r
e.g. i  Find the sum of the first 10 terms of 2  6  18  
a  2, r  3 and n  10 ar n  1
Sn 
r 1
2310  1
S10 
3 1
 59048

20. n1
1
8
ii   6 
n3  2 

21. n1 2
1 1
8
ii   6  a  6 
n3  2   2

22. n1 2
1 1
8
ii   6  a  6 
n3  2   2
3

2

23. n1 2
1 1
8
1
ii   6  a  6  r  ,n6
n3  2   2 2
3

2

24. n1 2
1 1
8
1
ii   6  a  6  r  ,n6
n3  2   2 2
3

2
a1  r n 
Sn 
1 r

25. n1 2
1 1
8
1
ii   6  a  6  r  ,n6
n3  2   2 2
3

3 1 
6
2 1    
a1  r n  2  2 
Sn  S6   
1 r 1
1
2

26. n1 2
1 1
8
1
ii   6  a  6  r  ,n6
n3  2   2 2
3

3 1 
6
2 1    
a1  r n  2  2 
Sn  S6   
1 r 1
1
2
3 63 2
  
2 64 1
189

64

27. n1 2
1 1
8
1
ii   6  a  6  r  ,n6
n3  2   2 2
3

3 1 
6
2 1    
a1  r 
n
2  2 
Sn  S6   
1 r 1
1
2
3 63 2
  
2 64 1
189

64
2
iii  Does 56  4    have a limiting sum?
7

28. n1 2
1 1
8
1
ii   6  a  6  r  ,n6
n3  2   2 2
3

3 1 
6
2 1    
a1  r 
n
2  2 
Sn  S6   
1 r 1
1
2
3 63 2
  
2 64 1
189

64
2
iii  Does 56  4    have a limiting sum?
7
4
r  1
56

29. n1 2
1 1
8
1
ii   6  a  6  r  ,n6
n3  2   2 2
3

3 1 
6
2 1    
a1  r 
n
2  2 
Sn  S6   
1 r 1
1
2
3 63 2
  
2 64 1
189

64
2
iii  Does 56  4    have a limiting sum?
7
4
r  1
56
as r  1, it has a limiting sum

30. 1 1 1
iv     
2 4 8

31. 1 1 1
iv     
2 4 8
1 1
a  ,r 
2 2

32. a
1 1 1
iv      S 
2 4 8 1 r
1 1
a  ,r 
2 2

33. a
1 1 1
iv      S 
2 4 8 1 r
1 1 1
a  ,r 
2 2  2
1
1
2
1

34. a
1 1 1
iv      S 
2 4 8 1 r
1 1 1
a  ,r 
2 2  2
1
1
2
1
v  Write 0.36 as a fraction


35. a
1 1 1
iv      S 
2 4 8 1 r
1 1 1
a  ,r 
2 2  2
1
1
2
1
v  Write 0.36 as a fraction


0.36  0.36  0.0036  0.000036  

36. a
1 1 1
iv      S 
2 4 8 1 r
1 1 1
a  ,r 
2 2  2
1
1
2
1
v  Write 0.36 as a fraction


0.36  0.36  0.0036  0.000036  
a  0.36, r  0.01

37. a
1 1 1
iv      S 
2 4 8 1 r
1 1 1
a  ,r 
2 2  2
1
1
2
1
v  Write 0.36 as a fraction


0.36  0.36  0.0036  0.000036  
a  0.36, r  0.01
a
S 
1 r

38. a
1 1 1
iv      S 
2 4 8 1 r
1 1 1
a  ,r 
2 2  2
1
1
2
1
v  Write 0.36 as a fraction


0.36  0.36  0.0036  0.000036  
a  0.36, r  0.01
a
S 
1 r
0.36

1  0.01
36

99
4

11

39. a
1 1 1
iv      S 
2 4 8 1 r
1 1 1
a  ,r 
2 2  2
1
1
2 Exercise 6J; 3cf, 5, 7b
1 8a(i), 9, 12, 14, 17b
v  Write 0.36 as a fraction


0.36  0.36  0.0036  0.000036   Exercise 6K; 2adgj, 3b,
a  0.36, r  0.01 4a, 6bf, 8, 11a, 17ace, 19*
a
S 
1 r Exercise 6L, 1dg, 2d, 4a
0.36

1  0.01
36

99
4

11