Graphics Reference
In-Depth Information
n
r
n
r
n
1
r
, when 0
r
n
.
1
n
r
n
!
Proveby induction that
r
)!
, taking 0!
1.
r
!(
n
n
r
n
n
r
Verify that this formula gives
.
What aspect of Pascal's triangle does this reflect?
5
The Binomial Theorem for positive integral index
Proveby induction that
n
0
n
1
n
2
n
r
n
n
(1
x
)
x
x
...
x
...
x
.
6
Examineeach of thepropositions:
(i) 2
n
,
(ii)
n
n
41 is a primenumbr.
Find values of
n
for which these propositions hold and also a value
of
n
for which each is false.
If you were to attempt a proof by induction of either of these
propositions, where would the proof break down?
7
Examinethenumbrs 6
5
n
1 and 6
5(
n
1)
1.
Find their difference.
Deduce that if the first of these numbers were divisible by 5 then
the second would also be divisible by 5. Deduce also that if the
second were divisible by 5 then the first would also bedivisibleby
5.
Is the first number divisible by 5 when
n
1?
Are there any values of
n
for which these numbers are divisible by
5?
If you were to attempt a proof by induction that the first number
was divisible by 5, where would the proof break down?
dy
dx
, and
y
dy
dx
.
8
Define
y
y
,
y
x
), so
dy
1
(i) Let
y
log
(1
dx
x
.
1
Provethat for
n
1, (1
x
)
y
ny
0.
Deduce that when
x
0,
y
(
1)
n
!
