14
NON-FOUNDATION
Arithmetic and Geometric
Sequences and their Summation
Name :
14E
Date :
Mark :
14.3B Geometric Mean
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Key Concepts and Formulae
If x, y and z are any three consecutive terms of a geometric sequence, then
y = ± xz .
1.
(a) Insert a positive geometric mean between 24 and 6.
1
2
(b) Insert a negative geometric mean between -18 and - .
(c) Insert a geometric mean between 7 and 35, leave your answer in surd form.
Solution
(a) Let x be the positive geometric mean between 24 and 6.
\
x =
( 24 )( 6 )
= ( 144 )
= ( 12 )
\
The required geometric mean is ( 12 ).
1
2
(b) Let x be the negative geometric mean between -18 and - .
\
x = - (-18)Ê - 1 ˆ
Ë 2¯
= - 9
= -3
\ The required geometric mean is -3.
(c) Let x be the geometric mean between 7 and 35.
\
x = ± 7(35)
= ± 245
= ±7 5
\ The required geometric mean is 7 5 or -7 5 .
95
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Number and Algebra
2.
If the geometric mean between 25 and y is -10, find the value of y.
Solution
Q
-10 is the geometric mean between 25 and y.
\
-10 = - ( 25 )( y )
(-10)2 = (- 25y )2
100 = 25y
y = ( 4 )
3.
(a) Insert two geometric means between 3 and 192.
4
and 36.
9
1
9
and - .
(c) Insert four geometric means between
27
32
(b) Insert three geometric means between
Solution
(a) Let R1 be the common ratio of the geometric sequence formed. The geometric sequence
formed is:
3, 3R1, ( 3R12 ), 192
Q
The 4th term is also given by 3R1(
\
3 R1
( 3
)
= 192
( 3
)
= ( 64 )
R1
3 )
.
R1 = ( 4 )
\
The two required geometric means are ( 12 ) and ( 48 ).
(b) Let R2 be the common ratio of the geometric sequence formed. The geometric sequence
formed is:
4
,(
9
), (
2
4
R2
9
), (
Q
The 5th term is also given by (
\
4 4
R 2 = 36
9
R2
96
4
R2
9
4
= 81
3
4
R2
9
4
4
R2
9
), 36
).
14
Arithmetic and Geometric Sequences and their Summation
R2 = ( 3 ) or ( -3 )
When R2 = ( 3 ), the three required geometric means are (
When R2 = ( -3 ), the three required geometric means are ( -
4
3
4
3
), ( 4 ) and ( 12 ).
), ( 4 ) and ( -12 ).
(c) Let R3 be the common ratio of the geometric sequence formed. The geometric sequence
formed is:
1
27
,
1
27
R3,
1
27
1
2
R3 ,
27
3
R3 ,
1
27
Q The 6th term is also given by
\
4
R3 , 1
27
9
32
5
R3 .
5
1
9
R3 = 27
32
R3
5
= -
R3 = -
243
32
3
2
\ The four required geometric means are -
4.
1
3
1
1
,
, - and
.
16
18 12
8
If x - 6, x + 2 and -5x + 2 are in geometeric sequence, find the values of x.
Solution
Q
x - 6, x + 2 and -5x + 2 are in geometric sequence.
\
( x+2 ) = ± ( x-6
)( -5x + 2 )
(x + 2)2 = [± (x - 6)(-5x + 2)]2
x 2 + 4x + 4 = -5x 2 + 32x - 12
6x 2 - 28x + 16 = 0
3x 2 - 14x + 8 = 0
(3x - 2)(x - 4) = 0
x =
2
or 4
3
97