Transcript Geometric sequences A sequence in which you get from one term to the next by multiplying by a constant is called a geometric.

Geometric sequences
A sequence in which you get from one term to the
next by multiplying by a constant is called a
geometric sequence.
This is also known as a geometric progression (GP)
and the constant multiplier is called the common
ratio.
 The first term of a GP is denoted by a.
 Its common ratio is denoted by r.
 Formula for the nth term of GP is arn-1
 nth term: un = arn-1
Examples
Decide which of the following sequences are
geometric progressions. If the sequence is GP then
write the common ratio and the next term.
(a) 3, 6, 12, 24, 48, 96
(b) 2, 2.4, 2.8, 3.2, 3.6, 4.00
(c) 3, 3.3, 3.63, 3.993, 4.3923
(d) 4, -12, 36, -108, 324, -972
(a) yes: r = 2,
Next term = 192
(b) no
(c) yes: r = 1.1, Next term = 4.83153
(d) yes: r = -3, Next term = 2916
Examples
Write down the term indicated in square bracket in
each of the following geometric sequences.
(a) 1, 2, 4, 8, 16,
[10th term]
(b) 5, -10, 20, -40, 80,
(c) 16, 8, 4, 2, 1,
[ 8th term]
[ 8th term]
(d) p, p3, p5, p7, p9, [ 9th term]
(a) a = 1, r = 2,
10th term =
1 x 29 =512
(b) a = 5, r = -2,
8th term =
(c) a = 16, r = ½ ,
8th term = 16 x ( ½ )7 = 1/8
(d) a = p, r = p2 ,
9th term = p x ( p2 )8 = p17
5 x (-2)7
= -640
Examples
Find an expression for the nth term of each of the
following GPs.
(a) 1, 2, 4, 8, 16, 32
(b) 5, -10, 20, -40, 80,
(c) 16, 8, 4, 2, 1,
(d) p, p3, p5, p7, p9,
(a) a = 1, r = 2,
nth term = 2n-1
(b) a = 5, r = -2,
nth term =
5 x (-2)n-1
(c) a = 16, r = ½ ,
nth term = 16 x ( ½ )n-1
(d) a = p, r = p2 ,
nth term = p x ( p2 )n-1
Examples
Find the number of terms in each of the following
GPs.
(a)
2, 10, 50, ….., 1250
(a) a = 2, r = 5
= 625
n–1=4
n =5
5, 20, 80, ……., 5120
(b) a = 5, r =
nth term = 2 x 5n- 1 = 1250
5n- 1 = 625
54
(b)
trial and
improve
ment
4
nth term = 5 x 4n- 1 = 5120
4n- 1 = 1024
trial and
improve
ment
45 = 1024
n–1=5
n =6
Examples
Find the common ratio and the first term in these GPs.
(a) the 2nd tem is 15 and the 5th term is 1875
(b) the 3rd term is 6 and the 7th term is 96
(a) 2nd term = ar = 15
[1]
5th term = ar4 = 1875
[2]
[2] [1] =
r3 = 125
Form [1] a x 5 = 15
(b) 3rd term = ar2 = 6
7th term = ar6 = 96
[2] [1] = r4 = 16
Form [1] a x 4 = 6
giving r = 5
giving a = 3
[1]
[2]
giving r = 2
giving a = 1.5