1.Algebra Booster

1.4 Algebra Booster
fi a + (n + 1)d = b
( b - a)
fi d =
n + 1
Êb
- aˆ
Hence, A1
= a + d = a + Á
Ë n + 1 ˜
¯ .
Similarly,
Êb-
aˆ
A2 = a + 2d = a + 2 Á
Ë n + 1 ˜
¯ .
o
Êb
- aˆ
An
= a + nd = a + n Á
Ë n + 1 ˜
¯
Property
If n arithmetic means inserted between two positive real numbers,
say a and b, the sum of the n-arithmetic means is equal
to n-times the single arithmetic mean between two positive
real numbers, i.e.
Êa
+ bˆ
A1+ A2+ A3+º+ An
= n¥Á Ë
˜
2 ¯ .
6.2 Geometric Mean
When three quantities are in GP, the middle one is called the
geometric mean (GM) between the other two.
If a, b and c are in GP, then b is the geometric mean between
a and c.
(i) If a, b ΠR + , then GM = (G) = ab .
(ii) If a, b, c ΠR + , then GM = (G) = 3 abc .
(iii) If a 1
, a 2
, a 3
, …, a n
ΠR + , then GM = (G) = n aa a .
1 2 …
Insertion of n geometric means between two
positive real numbers
Let two positive real numbers be a and b.
Let G 1
, G 2
, …, G n
are n geometric means inserted between
the two given positive numbers, a and b.
Then a, G 1
, G 2
, …, G n
, b must be in G.P.
Thus, t n–2
= b
fi ar n+1 = b
fi
Êbˆ
r = Á
Ë
˜
a¯
1
n+
1
1
1
Êbˆ
n+
So, G1
= ar = aÁ ˜ ,
Ë a ¯
Similarly,
2
2 Êbˆ
n+
1
G2
ar a a
o
= = Á
Ë
˜
¯
n
n Êbˆ
n+
1
Gn
ar a a
= = Á
Ë
˜
¯
n
Property
The product of n geometric means inserted between two positive
numbers, say a and b, is equal to the nth power of the
single geometric mean between the two given positive numbers,
i.e.
G 1
, G 2
, …, G n
= (ar)(ar 2 ) … (ar n )
= a n ¥ r 1+2+3+ … + n
n
nn ( + 1)
2
= a ¥ ( r)
È
nn ( + 1) ˘
Í 1+ 2+ 3+ + n =
Î
2
˙
˚
Ê
n Êbˆ
= a ¥ Á
Ë Á
Ë ˜
a¯
n
2
n Êbˆ
= a ¥ Á
Ë ˜
a¯
È Êbˆ
Í Á ˜
Î Ëa¯
n
= ( ab)
2
n
= ( ab) .
6.3 Harmonic Mean (HM)
nn ( + 1)
1
ˆ
2
n + 1
m
˜
¯
m
= 6 ¥ a
-m
When three non-zero quantities are in HP, the middle one is
called the harmonic mean (HM) between the other two.
If a, b and c are in HP, then b is called the harmonic mean
between a and c.
(i) If a, b ΠR + , then HM = (H) =
2
1 1
+
a b
3
(ii) If a, b, c ΠR + , then H = .
1 1 1
+ +
a b c
(iii) If a 1
, a 2
, a 3
, …, a n
ΠR + , then
n
H =
1 1 1 1
+ + +º+
a a a a
1 2 3
Insertion of n harmonic means between two
positive real numbers
Let two positive numbers be a and b respectively.
Let H 1
, H 2
, …, H n
are n harmonic means inserted between
two positive real numbers a and b.
Then a, H 1
, H 2
, …, H n
, b are in HP.
1 1 1 1 1
Thus, , , ,…, , are in AP.
a H H H b
1 2
Now 1 = 1 + ( n+
1) d
b a
n
n
˘
˙
˚