It
is Wise,
Generalize!
by K. R. S. Sastry,
Karnataka, India
Often
it is the case that the solution of one problem enables
us to propose several new problems. This may take the
form of a converse problem, an extension of the original
problem in one of several directions or an analogy. By
all means do consider them. It provides us with more problems
to solve on the one hand. On the other, we generally employ
different solution strategies to solve the newly proposed
problems so we gain greater perspectives and valuable
insights. Let us look at a solution of the following problem
in the light of preceding suggestions.
| (*)
Find a sequence of distinct natural numbers with the
property that the difference of squares of any two consecutive
terms is a perfect cube. |
|
Editor's
Note: Try the problem yourself before reading on!
Solution.
If we assume s ,
s ,
. . . as the sequence of natural numbers to be determined,
then we must have
s
- s
= t ,
r =
1, 2, 3, .....
|
(1) |
where
t's are also natural numbers. When r = 1 then
s
- s
= t .
A little experimentation shows that 3
- 1=
2 .
So we take s
= 1, s
= 3, t
= 2. When r = 2 then s
- s
= t
holds. With s
= 3, we find that s
= 6. Likewise we find that s
= 10. Let us write down the initial terms of the sequence:
1,
3, 6, 10, s ,
s ,
. . .
It
is now easy to guess that s
= 15, s
= 21, and that s
= r(r + 1)/2. By using the induction principle
we can convince ourselves that
s
= r(r + 1)/2 r
= 1, 2, 3, . . .
|
(2) |
is
the correct formula for a solution to our problem. It is
hoped that the reader will complete the induction step.
A
Pythagorean Concept
Hindsight
tells us that (*) was a relatively easy problem to solve.
If we think so and forget all about it then much of the
mathematical pleasure and the thrill of mathematical discovery
will be lost. After solving the problem (*) one question
we might ask ourselves is this: Does the sequence
|
1,
3, 6, 10, . . ., r(r + 1)/2, . . .
|
(3) |
have
any special significance? The formula (2) might give us
a clue: 1 = 1,
3 = 1+2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, . . . , r(r
+ 1)/2 = 1 + 2 + 3 + . . . + r, and so s
is the sum of the first r natural numbers for
r = 1, 2, 3, . . . Ancient Pythagoreans, students
of Pythagoras - the name attached to the famous theorem
on the squares of the sides of a right-angled triangle -
noticed the preceding property and more. They discovered
that they could form a triangle figure for each term of
(3) using that same number of objects. Here it goes: The
triangle figure for
 s
= 1 contains one object on each side. Here we say the
triangle shrinks to a point.
s
= 3 contains two objects on each side. The total number
of objects used is 3.
s
= 6 contains three objects on each side. The total number
of objects used is 6. Notice that this triangle figure is
an extension of the one for s.
s
= 10 contains four objects on each side. The total number
of objects now used is 10. Notice again that the present
triangle figure is an extension of the previous one for
s.
The
reader may depict a triangle figure using a total of 15
or 21 objects. The figure for s
contains r objects on each side and a total of
r(r + 1)/2 objects. So the ancient Pythagoreans
natrually called
1,
3, 6, 10, . . ., r(r + 1)/2, . . .
the
sequence of triangular numbers. For future reference we
restate the property observed earlier:
|
Each
triangular number is a sum of an arithmetic progression
whose first term is 1 and the common difference is
1.
|
(4) |
Generalization
to n-gonal Numbers
Curiosity
compels us to ask and answer a host of questions such as
(i)
What sequence of numbers might be called a sequence of square
numbers? The answer is built into the question itself: 1,
4, 9, 16, ..., r,
... Try to depict a sequence of square figures using a total
of 1, 4, 9, 16, ... objects respectively. Discover a property
of square numbers analogous to (4) in the solution of the
following problem:
| Find
the arithmetic progression whose sequence of sums is
the sequence of square numbers. (The sequence intended
is the one formed by taking the sum of the first 1 term,
then the first 2 terms, then the first 3 terms, etc.) |
The
answer is a well known property of positive odd integers.
Our discussion has provided a new meaning to it while placing
it in the natural context of development of the concept
of n-gonal numbers.
(ii)
What sequence of numbers might be called a sequence of pentagonal
numbers? The following pentagonal figures should help deciphering
the first few pentagonal numbers - simply count the total
number of objects in each figure.
Deduce
the formula P(5,r) = r(3r -
1)/2 for the rth pentagonal number. Find the arithmetic
progression whose sequence of sums is the sequence of pentagonal
numbers.
The
success with the triangular, the square, and the pentagonal
numbers should encourage us to find the sequence, the formula,
and the arithmetic progression whose sums will yield the
sequence in the case of hexagonal, heptagonal, octagonal,...,
n-gonal numbers, which are also called polygonal
numbers. This is one direction of thought that we pursued.
Are there others?
Further
Directions
Let us revisit problem (*) for further inspiration.
I.
We obtained the sequence of triangular numbers as a solution
to the problem (*). Can we find a different, finite or infinite
sequence of distinct natural numbers that is also a solution
to the problem (*)? Note that a subset of (3), finite or
infinite, such as {15,21,28,36} or {15,21,28,36,45,...}
cannot be considered a distinct solution to the problem
(*).
II.
We might also propose an analogous problem. Find a sequence
of distinct natural numbers such that the difference of
squares of any two consecutive terms is (a) a perfect square,
(b) a perfect biquadrate (fourth power).
| Finally,
try to pose and solve other variants of the problem
(*). Of course, we should be aware that posing such
a problem does not guarantee a solution to it. It is
the mathematical experience that matters most! |
|