Math 141 Homework #12
Due Tuesday, 11/13/07
Bonus Problem: The extremely important proof technique of mathematical induction is often used to
show that some fact is true for every positive integer. For example, the sums
S(n, p) =
n
X
i=1
ip
that come up in integration have closed-form formulas like
S(n, 1) =
n
X
i=1
i=n(n+ 1)
2,(1)
S(n, 2) =
n
X
i=1
i2=n(n+ 1)(2n+ 1)
6,(2)
S(n, 3) =
n
X
i=1
i3=n2(n+ 1)2
4.(3)
To prove that these formulas work for every positive integer n, you can use induction. Induction is vital in
many other areas of mathematics, including sequences and series (which you’ll see in Math 122/142) and
identities involving binomial coefficients (think back to the first few days of class).
The technique of induction can be a bit confusing at first, but with a little practice, you can get used to how
it works, and it is very much worth learning. A brief summary appears in the box on p.87 of the textbook,
although it’s not very enlightening by itself; you really have to work through a couple of examples to get the
hang of how the technique works. The Wikipedia article on induction at
http://en.wikipedia.org/wiki/Mathematical induction
has a more detailed description of induction that is readable and accurate, including a proof of the formula
(1) (which is the “standard” example of induction). The formula (2) is proved inductively in Appendix F
on p. A47. A slightly different example of induction appears on p. 90.
On the other hand, it is possible to prove formulas like (1), (2) and (3) without induction, as in Examples 4
and 5 on pp. A46–A47.
Read all this material. Once you have done so, mimic the method of Example 5 to find a formula for S(n, 4),
and check that it works for several values of n(say 1 ≤n≤5). (If you want, you can check your formula
for S(n, 4) against the Wikipedia article on Faulhaber’s formula; see below.)
Next, give another proof of your formula by induction.
Finally, find a recursive formula for S(n+ 1, p) in terms of S(n, p), again by mimicking the method of
Example 5. (Hint: In order to make the method work for all n, you will need binomial coefficients.)
For the curious, there is a general formula for S(n, 4) called Faulhaber’s formula; see
http://en.wikipedia.org/wiki/Faulhaber’s formula.
However, the formula involves things called Bernoulli numbers, which are not easy to write in closed form;
indeed, to give a general formula for them, you need mathematical induction again!