The document defines a geometric series as a sequence where each term is found by multiplying the previous term by a constant ratio. It provides the following key points: - The constant ratio is called the common ratio (r) - The general term (Tn) is equal to the first term (a) multiplied by the common ratio raised to the power of n-1 - Worked examples are provided to find the common ratio and general term given values in a sequence, as well as to determine the first term greater than a specified value.

1. Geometric Series

2. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.

3. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.

4. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T
r 2
a

5. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T
r 2
a
T
 3
T2

6. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T
r 2
a
T
 3
T2
Tn
r
Tn1

7. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1  a
r 2
a
T
 3
T2
Tn
r
Tn1

8. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1  a
r 2
a T2  ar
T
 3
T2
Tn
r
Tn1

9. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1  a
r 2
a T2  ar

T3 T3  ar 2
T2
Tn
r
Tn1

10. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1  a
r 2
a T2  ar

T3 T3  ar 2
T2 Tn  ar n1
Tn
r
Tn1

11. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1  a
r 2
a T2  ar

T3 T3  ar 2
T2 Tn  ar n1
Tn
r
Tn1 e.g.i  Find r and the general term of 2, 8, 32, 

12. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1  a
r 2
a T2  ar

T3 T3  ar 2
T2 Tn  ar n1
Tn
r
Tn1 e.g.i  Find r and the general term of 2, 8, 32, 
a  2, r  4

13. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1  a
r 2
a T2  ar

T3 T3  ar 2
T2 Tn  ar n1
Tn
r
Tn1 e.g.i  Find r and the general term of 2, 8, 32, 
Tn  ar n1 a  2, r  4

14. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1  a
r 2
a T2  ar

T3 T3  ar 2
T2 Tn  ar n1
Tn
r
Tn1 e.g.i  Find r and the general term of 2, 8, 32, 
Tn  ar n1 a  2, r  4
 24
n1

15. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1  a
r 2
a T2  ar

T3 T3  ar 2
T2 Tn  ar n1
Tn
r
Tn1 e.g.i  Find r and the general term of 2, 8, 32, 
Tn  ar n1 a  2, r  4
 24
n1
 22 
2 n 1
 22 
2 n2

16. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1  a
r 2
a T2  ar

T3 T3  ar 2
T2 Tn  ar n1
Tn
r
Tn1 e.g.i  Find r and the general term of 2, 8, 32, 
Tn  ar n1 a  2, r  4
 24
n1
 22 
2 n 1
Tn  22 n1
 22 
2 n2

17. ii  If T2  7 and T4  49, find r

18. ii  If T2  7 and T4  49, find r
ar  7

19. ii  If T2  7 and T4  49, find r
ar  7
ar 3  49

20. ii  If T2  7 and T4  49, find r
ar  7
ar 3  49
r2  7

21. ii  If T2  7 and T4  49, find r
ar  7
ar 3  49
r2  7
r 7

22. ii  If T2  7 and T4  49, find r (iii) find the first term of 1, 4, 16, … to
ar  7 be greater than 500.
ar 3  49
r2  7
r 7

23. ii  If T2  7 and T4  49, find r (iii) find the first term of 1, 4, 16, … to
ar  7 be greater than 500.
ar 3  49 a  1, r  4
r2  7
r 7

24. ii  If T2  7 and T4  49, find r (iii) find the first term of 1, 4, 16, … to
ar  7 be greater than 500.
Tn  14
n1
ar 3  49 a  1, r  4
r2  7
r 7

25. ii  If T2  7 and T4  49, find r (iii) find the first term of 1, 4, 16, … to
ar  7 be greater than 500.
Tn  14
n1
ar 3  49 a  1, r  4
r2  7 Tn  500
r 7

26. ii  If T2  7 and T4  49, find r (iii) find the first term of 1, 4, 16, … to
ar  7 be greater than 500.
Tn  14
n1
ar 3  49 a  1, r  4
r2  7 Tn  500
r 7 4n1  500

27. ii  If T2  7 and T4  49, find r (iii) find the first term of 1, 4, 16, … to
ar  7 be greater than 500.
Tn  14
n1
ar 3  49 a  1, r  4
r2  7 Tn  500
r 7 4n1  500
log 4n1  log 500

28. ii  If T2  7 and T4  49, find r (iii) find the first term of 1, 4, 16, … to
ar  7 be greater than 500.
Tn  14
n1
ar 3  49 a  1, r  4
r2  7 Tn  500
r 7 4n1  500
log 4n1  log 500
n  1log 4  log 500

29. ii  If T2  7 and T4  49, find r (iii) find the first term of 1, 4, 16, … to
ar  7 be greater than 500.
Tn  14
n1
ar 3  49 a  1, r  4
r2  7 Tn  500
r 7 4n1  500
log 4n1  log 500
n  1log 4  log 500
n  1  4.48
n  5.48

30. ii  If T2  7 and T4  49, find r (iii) find the first term of 1, 4, 16, … to
ar  7 be greater than 500.
Tn  14
n1
ar 3  49 a  1, r  4
r2  7 Tn  500
r 7 4n1  500
log 4n1  log 500
n  1log 4  log 500
n  1  4.48
n  5.48
T6  1024, is the first term  500

31. ii  If T2  7 and T4  49, find r (iii) find the first term of 1, 4, 16, … to
ar  7 be greater than 500.
Tn  14
n1
ar 3  49 a  1, r  4
r2  7 Tn  500
r 7 4n1  500
log 4n1  log 500
n  1log 4  log 500
n  1  4.48
n  5.48
T6  1024, is the first term  500
Exercise 6E; 1be, 2cf, 3ad, 5ac, 6c, 8bd, 9ac, 10ac, 15, 17,
18ab, 20a