Prove It

After observing that

               1  =  1
and         1 + 3  =  4
and         1 + 3 + 5  =  9
and         1 + 3 + 5 + 7  =  16
and         1 + 3 + 5 + 7 + 9  =  25

it seems easy to conclude that, for any positive integer n, the sum of the first n odd integers is n².

Or if we observe that

                (11)¹  =  11
and          (11)²  =  121
and          (11)³  =  1331
and          (11)⁴  =  14741

then we may want to conclude that all positive integer powers of 11 are palindromes.  But this is not so, since  (11)⁵ = 162151.

To test a rule or formula for any finite number of cases, calculation is possible.  But for an infinite number of cases (such as all positive integers) complete results can never be calculated and a logical argument or proof is required.  For maturing math students, initial efforts to "prove it" are challenging and sometimes frustrating. 

Last week while browsing in the collection, Painted Bride Poetry:  A Poetry Retrospective 1973-1993, I found the following poem, "Prove It" by Nebraska poet and teacher William Kloefkorn (1932-2011).  While Kloefkorn's poem does not refer at all to mathematics, it captures the frustration encountered in trying to prove something.  And the difference between knowing and proving.  (Such as in the first example above involving the sum of odd integers.)

          Prove It     by  William Kloefkorn

          I see Bubba Barnes
          sneak a comic book
          from the rack in
          the Rexall drug-
          store, and the next
          day at recess
          I tell him.  He 
          says Prove it.

          I even saw the
          name of the comic,
          I tell him.  Sub-
          mariner. Isn't 
          that right?  He
          says Prove it.

          I don't have to
          prove it, I ay.
          I know you did it
          and you know you
          did it.  So, he 
          says, prove it, ass-
          eyes.  Just prove it.

          You can go to
          hell for swearing,
          I say.  Bubba says
          Prove it.  And for
          stealing, I say,
          and for not tell-
          ing the truth.  Bub-
          ba says Prove it
          prove it prove
          it prove it
          prove it.