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In-Depth Information
with thenotation of qn 46.
48
1
n
1
and
a
1
1
n
Put
b
1
in the left-hand inequality of qn 43 to prove that
b
b
(with
thenotation of qn 46).
49 Justify the inequalities
a
a
b
b
, with thenotation of
44 and 46, and use46 to show that
a
3 for all positive integers
n
. (This result will be used in qn 4.36.)
n
th roots
50 If 1
a
, provethat
a
a
a
(i) 1
...
,
(ii) 1
a
a
...
a
1
a
a
...
a
a
,
n
n
1
(iii)
a
1
n
a
1
n
1
(from qn 1.3(vi)),
(iv) if
a
b
,(
n
1)(
b
1)
n
(
b
1).
51 Deduce from qn 49 that if
n
3, then (1
1/
n
)
n
, and so
n
.
(
n
1)
n
n
1.5
1
0.5
n
1
2
3
4
5
6
7
8
9
10
Summary
-
results on inequalities
Arithmetic and geometric means
If
a
and
b
are positive, then
(
ab
)
(
a
b
).
