1.Algebra Booster

Sequence and Series 1.5
fi
fi
Therefore,
Similarly,
1 1
- = ( n+
1) d
b a
a - b
d =
( n+
1) ab
1 1 1 Ê a - b ˆ
= + d = +
H1
a a Á
Ë( n+
1) ab ˜
¯
1 1 1 Ê a - b ˆ
= + d = + 2
H2
a a Á
Ë ( n+
1) ab ˜
¯
1 1 1 Ê a - b ˆ
= + d = + 3
H a a Á
Ë ( n+ 1) ab ˜
¯
3
o
1 1 1 Ê a - b ˆ
= + d = + n
H a a Á
Ë( n+
1) ab ˜
¯
n
Property
The sum of the reciprocal of n harmonic means is equal to n
times the single harmonic mean between the two given positive
real numbers.
Let H 1
, H 2
, …, H n
are n harmonic means inserted between
two positive real numbers a and b.
Since a, H 1
, H 2
, …, H n
, b are in HP
1 1 1 1 1
, , ,…, , are in AP
a H1 H2
Hn
b
Ê 1 1ˆ
1 1 1 +
Therefore, + + … +
=
a b
n ¥
H1 H2
H
Á
n
Ë
˜
2 ¯
1
= n ¥ Ê 2 ˆ
Á 1 1˜
Á +
Ë
˜
a b¯
Relation amongst the AM, GM and HM
Let a, b ΠR + .
Êa
+ bˆ
Then AM = (A) = Á
Ë
˜
2 ¯ ,
GM = (G) = ab
Ê 2ab ˆ
and HM = (H) = Á
Ëa
+ b
˜
¯
Êa + bˆ Ê 2ab
ˆ
Now, A ¥ H = Á ¥
Ë
˜
2 ¯ Á
Ëa
+ b
˜
¯
= ab = G 2
Thus, A, G, H are in GP.
fi G 2 = AH
fi
A G
=
G H
…(i)
a + b
Also, A – G = - ab
2
1 ( )
2
= a - b ≥ 0
2
A
fi A≥Gfi ≥1
G
From Relations (i) and (ii), we get
G A
= ≥1
H G
fi G ≥ H
Hence, A ≥ G ≥ H
fi AM ≥ GM ≥ HM
…(ii)
Notes
(i) If
n+ 1 n+
1
a + b
n n
a + b
be the AM between two numbers a
and b, then n = 0
n+ 1 n+
1
a + b
(ii) If
be GM between two positive numbers
n n
a + b
a and b, then n = –1/2
n+ 1 n+
1
a + b
(iii) If
be the HM between two positive
n n
a + b
numbers a and b, then n = –1.
7. mth POWERS THEOREM
If a 1
, a 2
, …, a n
be a set of positive numbers and all the a’s are
not equal,
m
Ê n ˆ n
m Ê ˆ
ÁÂ
ai
˜ ÁÂ
ai
˜
Ë i= 1 ¯ Á i=
1
> ˜ ,
n Ë n ¯
when 0 < m < 1
m
Ê n ˆ n
m Ê ˆ
ÁÂ
ai
˜ ÁÂ
ai
˜
Ë i= 1 ¯ i=
1
and
< Á ˜
n Ë n ¯
when m ΠR Р(0, 1).
8. CAUCHY-SCHWARTZ INEQUALITY
If a, b, c and x, y, z are any real numbers (positive, negative
or zero), then
(a 2 + b 2 + c 2 )(x 2 + y 2 + z 2 ) ≥ (ax + by + cz) 2
9. MAXIMUM AND MINIMUM VALUES OF
POSITIVE REAL NUMBERS
Let us suppose that x, y, z, …, w and are n positive variables
and c is constant.