1oy T T T TR ST A

R PR

4 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

the nth term of the series of natural numbers is #, the nth
term of the series of squares of the natural numbers is #?, the
nth even number starting with 2 is 2%, the nth odd number
starting with 1 is 2z — 1, the nth odd number starting with
3is 2n + 1.

13. Let us see how the principle of mathematical induction
can be used to prove such statements.

Theorem. First let us prove that the nth even number start-
ing with 2 is 2n, that is, that, if a, represents this nth even
number, then @, = 2n, or that the statement a, = 2n is tru
for all valuesof #,or thata; = 2 X 1,a:, = 2 X 2,03 = 2 X 3,
Cigy e 2 e

Here we have a series of statements of which we wish to
prove that all are tru.

The first statement of the series, a; = 2 X 1, is evidently
tru by hypothesis.

Assume that the kth statement is tru, that is, that

(Lk=2k

Now @iy = ar + 2, as any even number can be obtaind
from the preceding even number by adding 2 to that number.
But, since we assumed that a; = 2k,

ar+2=2k+2=20Fk+1)

Hence
Ar41 = Z(k + 1)

Thus the kth statement, @, = 2k, in our series of statements
implies the 2 4 1th statement, ar = 2(k + 1).

Therfor the statements are all tru.

That is, the nth term in the series of even numbers starting
with 2 is 2n.

14. Th. Next let us prove that the nth odd number start-
ing with 1 is 2n — 1, that is, that, if @, represents now this
nth odd number, then ¢, =2 X1—-1, ae =2 X2 — 1,
(l:;:2 ><3— 1, "',(Lk=2k— 1, e