14 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

The three series whose nth terms are 72, 2n? — 3n + 2, and
n® — 5n? + 11n — 6 respectivly have the same first two terms;
the first and last of these have the same first three terms.

If a finite number of terms at the beginning of a series are
given, therfor, and nothing more, the n#th term cannot be
found, that is, the series is not determind. (%)

37. It is plain that if the first of a series of statements is
tru they may not all be tru. That is, the first clause in the
hypothesis of the principle of mathematical induction does not
imply the conclusion.

Moreover the second clause does not imply the conclusion,
as may be seen by a simple example.

Suppose that we were trying to prove that the nth even
number starting with 2 is 2» + 5. Letting a, stand for this
nth even number, and supposing that a, = 2k + 5, we would
have as a consequence

(lk+1=ak+2=2k+5+2=2(k+1)+5
Hence a1 =2k+1)+4+5

So that the kth statement in this series of statements would
imply the & + 1th; if the kth statement were tru, so would
be the 2 + 1th. But none of them are tru.

As another example, suppose we assumed that the sum of
the first £ odd numbers starting with 1 was 3 4 %%, that is,
that Sy = 3 + k2. Then, setting a; for the kth odd number
starting with 1, it would follow that

Sk+1=Sk+ak+1=3+k2+(2k+1)=3+(k+1)2

Here also the kth statement would imply the 2 4+ 1th. But
none of them are tru.

38. We may also remark that the method of mathematical
induction gives us no clue to a formula for the nth term of a

(1) In spite of this fact however text books on algebra frequently give
the first three or four terms of a series and expect the student to find the
nth term,