Pascal’s Triangle, Induction and the Binomial Theorem
Induction:
a.
Suppose that the only currency were 3-Euro bills and 10-Euro notes. Show
that any amount greater than Euro 17 could be made from a combination of
these notes.
b.
Prove that the following equality holds for every
c.
Prove that
d.
Show that
e.
Prove that
f.
Show that 21 divides
g.
Prove that
h.
Let
for all
is divisible by 5 for all
.
for any positive integer n.
is divisible by 4 for all
.
be the terms of a sequence defined by:
Prove that each
.
is divisible by 3.
Prove that
is a multiple of 6 for all natural numbers .
Pascal’s Triangle:
Using the Hockey Stick formula or in any other way, find a formula for:
a)
b)
c)
d)
e)
f)
g)
.
for all
and
j.
:
.
..
Can you explain/prove the following patterns in Pascal’s
Triangle?
Pattern 1:
Left-Right symmetry.
Pattern 2:
Hockey Stick Formulas (see diagram).
Pattern 3:
Sum of elements on a horizontal line.
Hint: Use the Binomial formula for a special value of X.
Pattern 4a:
Sum of elements on a horizontal line, with
alternate signs.
Hint: Use the Binomial formula for a special value of X.
Pattern 4b:
Sum of elements on a horizontal line,
skipping by twos.
.
Activity:
In Pascal’s Triangle, colour all the odd numbers black, and leave the even numbers white. Do
you notice a pattern? Can you explain it?
Look up the Sierpinski Triangle for more.
Binomial formula exercises:
Use the binomial expansion (or anything else you like) to solve these:
a.
Expand
.
b.
If you expand
c.
If you expand
d.
If you expand
e.
What is the coefficient of
in
f.
What is the coefficient of
of
g.
What is the highest power of
what will the coefficient of
what will the coefficient of
be?
be?
is there a term independent of ? If so, what is it?
Can you find its coefficient?
?
in
in
?