Graphics Reference
In-Depth Information
29 Result trivial for
n
1.
1
x
0
1
x
,so1
nx
(1
x
)
(1
nx
)(1
x
)
(1
x
)
1
(
n
1)
x
nx
(1
x
)
1
(
n
1)
x
(1
x
)
.
30
A
5,
G
4.
31
A
2
,
G
6. (2
)
6. Also 2
6
5.
32 0
(
a
b
)
2
ab
a
b
. Equality only when
a
b
.
33 0
(
a
b
)
2
a
b
a
b
a
b
(
a
b
). Equality only
when
a
b
.
34 Put
b
1/
a
in qn 33.
35 If thearms of thescaleare
l
(on thelft) and
r
(on theright), and the
two amounts of apples weighed out are
v
(on thelft) and
w
(on the
right),
lv
r
· 1 and
rw
l
· 1.
Now
v
w
r
/
l
l
/
r
2, from qn 34 unless
l
r
.
36 If the sides of the rectangle are of length
a
and
b
, theconstant
perimeter
2(
a
b
), whilethearea is
ab
.
ab
(
a
b
) which is
constant, so
ab
and thus the area is maximum when
a
b
.
37
G
a
,
A
(
x
a
/
x
). GM
AM
a
x
x
, except possibly
for
n
1. This is the basis of Heron's method for finding square roots.
38
AM and induction.
(iii) needs (ii) and induction.
(v)
b
(i) needs GM
a
(
b
a
)
.
Such pairs of sequences were investigated by Gauss.
39 Both
a
and
b
approach 1/2
as
n
increases.
40
(i)
c
9,
(ii)
d
(3/2)
,
(iii)
e
(
b
/2)
,
(iv)
f
b
/2,
x
(
f
(
f
ac
))/
a
(
b
(
b
4
ac
))/2
a
.
41 From theidentity,
a
x
abx
ac
ac
b
. Since
a
0,
ax
bx
c
(
ac
b
)/
a
. Equality is possibleonly when
0.
42
ax
bx
c
ax
b
(
ax
b
)
/
a
(4
ac
b
)/4
a
. For 0
a
, theright-hand
sideis positiveor 0 for all
x
precisely when 4
ac
b
is positiveor
zero. If
b
4
ac
, theright-hand sideis always positiv, and so
cannot
0.
For
a
0,
ax
bx
c
0 provided
b
4
ac
.
44
a
2.25,
a
2.37,
a
2.44,
a
2.49.
46
b
3.37,
b
3.16,
b
3.05,
b
2.99.
