Graphics Reference
In-Depth Information
b
a
(
a
b
) and
b
(
a
),
provethat
(i) 0
a
a
b
b
,
(ii) 0
b
a
b
a
(
b
a
),
(iii) 0
b
a
(
)
(
b
a
).
So the two means get closer together as
n
increases.
(iv) Provethat
b
a
(
b
a
)
and that when
1
a
b
,
b
a
(
b
a
).
(v) If
a
1 and
b
2, show that
b
a
1/2
, so thetwo
sequences close in very fast.
39 (optional) If a regular
n
-gon is inscribed in a circle of unit radius,
each side has length 2 sin
/
n
, so its perimeter is 2
n
sin
/
n
. Denote
this perimeter by
I
. If a regular
n
-gon is circumscribed about a
circle of unit radius, each side has length 2 tan
/
n
, so the perimeter
is 2
n
tan
/
n
. Denote this perimeter by
C
.
2
I
C
Provethat
I
(
I
C
) and that
C
.
I
C
1
C
1
Let
a
and
b
.
I
Deduce that
a
(
a
b
) and
b
(
b
a
).
a
a
b
b
Thus 0
.
Provethat
b
a
(
b
a
), so the two sequences close in
on each other. Check that
a
and
b
1/(4
2). What number
do you think
a
approach as
n
increases? (If you run a
computer program which gives successive values of
a
and
b
and
b
you
may be surprised to see that the terms of the two sequences
approach each other much more slowly than the terms of the
sequences in qn 38.)
Completing the square
40
(i) For what valueof
c
is
x
6
x
c
a perfect square?
(ii) For what valueof
d
is 4
x
6
x
d
a perfect square?
(iii) What expression for
e
will make
a
x
abx
e
a perfect
square?
(iv) Assuming that
a
0, find a formula for
f
such that the
equation
ax
bx
c
0 is equivalent to the equation
(
ax
f
)
f
ac
. Solvethis equation for
x
, when
f
ac
.
