Graphics Reference
In-Depth Information
Check this proposition for
n
1, 2 and 3.
By examining the difference between this number and
6
4, show that if theproposition holds for one
valueof
n
, it holds for the succeeding value of
n
. When you have
done this you have established the two components of the proof of
theproposition by mathematical induction.
5(
n
1)
3
Provethefollowing propositions by induction (somehaveeasy
alternative proofs which do not use induction):
(i) 1
2
3
...
n
n
(
n
1),
(ii)
a
(
a
d
)
(
a
2
d
)
...
(
a
(
n
1)
d
)
n
(2
a
(
n
1)
d
),
(iii) 1
2
3
...
n
[
n
(
n
1)]
,
(iv) 1 · 2
2· 3
3 · 4
...
n
(
n
1)
n
(
n
1)(
n
2),
(v)
2
4
...
2
2
1,
x
1
(vi) 1
x
x
...
x
1
, provided
x
1.
x
nx
x
x
1
(vii) (optional) 1
2
x
3
x
...
nx
1
,
(
x
1)
provided
x
1.
4
Pascal's triangle, shown here, is defined inductively, each entry
being the sum of the two (or one) entries in the preceding row
nearest to the new entry. The (
r
1)th entry in the
n
th row is
denoted by (
)
11
121
1331
14641
1
5
10
10
5
1
1
...
...
...
...
...
1
and called '
n
choose
r
' because it happens to count the number of
ways of choosing
r
objects from a set of
n
objects. Notice that we
always have0
r
n
. From theportion of Pascal's trianglewhich
has been shown, for example,
3
0
3
1
3
2
3
3
1,
3,
3 and
1.
From thedefinition of Pascal's trianglewehave
