an inequality usually known as Schwarz’s (though due originally to Cauchy).
11. If
a
1
,
a
2
, . . . ,
a
n
are all positive, and
s
n
=
a
1
+
a
2
+
· · ·
+
a
n
, then
(1 +
a
1
)(1 +
a
2
)
. . .
(1 +
a
n
)
5
1 +
s
n
+
s
2
n
2!
+
· · ·
+
s
n
n
n
!
.
(
Math. Trip.
1909.)
12. If
a
1
,
a
2
, . . . ,
a
n
and
b
1
,
b
2
, . . . ,
b
n
are two sets of positive numbers,
arranged in descending order of magnitude, then
(
a
1
+
a
2
+
· · ·
+
a
n
)(
b
1
+
b
2
+
· · ·
+
b
n
)
5
n
(
a
1
b
1
+
a
2
b
2
+
· · ·
+
a
n
b
n
)
.
13. If
a
,
b
,
c
, . . .
k
and
A
,
B
,
C
, . . .
K
are two sets of numbers, and all of
the first set are positive, then
aA
+
bB
+
· · ·
+
kK
a
+
b
+
· · ·
+
k
lies between the algebraically least and greatest of
A
,
B
, . . . ,
K
.
14. If
√
p
,
√
q
are dissimilar surds, and
a
+
b
√
p
+
c
√
q
+
d
√
pq
= 0, where
a
,
b
,
c
,
d
are rational, then
a
= 0,
b
= 0,
c
= 0,
d
= 0.
[Express
√
p
in the form
M
+
N
√
q
, where
M
and
N
are rational, and apply
the theorem of
§
14
.]
15. Show that if
a
√
2 +
b
√
3 +
c
√
5 = 0, where
a
,
b
,
c
are rational numbers,
then
a
= 0,
b
= 0,
c
= 0.
16. Any polynomial in
√
p
and
√
q
, with rational coefficients (
i.e.
any sum of
a finite number of terms of the form
A
(
√
p
)
m
(
√
q
)
n
, where
m
and
n
are integers,
and
A
rational), can be expressed in the form
a
+
b
√
p
+
c
√
q
+
d
√
pq,