SERIES AND MATHEMATICAL INDUCTION 3

the series of even numbers 2, 4, 6, 8, 10, ---;
the series of prime numbers 1, 2, 3,5, 7, 11, - --.

8. It should be remarkt that series may agree in any finite
number of terms at the beginning and differ in their other
terms. Thus the series of natural numbers, the series of the
squares of the natural numbers, the series of odd numbers,
and the series of prime numbers all begin with 1. The series
of natural numbers and the series of prime numbers both
begin with 1, 2, 3.

9. For a series to be given some rule must be given by which
any one of its terms can be found; by the rule we must be
able to find the 1st term, the 2d term, the 3d term, etc.; in
other words, given 7, it must be possible to find the nth term.

The nth term of a given series is then uniquely determind
when 7 is given, that is, it may be obtaind from # by some
univalent operation, of which it is the result. This result is
said to be a function of 7z, which simply means that the result
is obtaind from # by an operation.

If the terms of a series are denoted by ai, as, a3, ---, the
nth term is denoted by @,, a symbol which indicates that this
term is a function of .

10. If we let sy, S2, $3, * -+, Sn, + -+ stand for an infinit series
of statements, we may state the principle of mathematical
induction for such a series in symbols as follows:

[si= (G D s s

11. If our explanations have been understood, it must now
be clear that if the first of a series of statements is tru, and,
if any particular statement is tru and has a follower, the
follower is tru also, then they are all tru. Since the first is
tru, the second is; since the second is tru, so is the third; since
the third is tru, so is the fourth; and so on.

12. Sometimes the nth term of a series can be exprest in
terms of # by means of well known operations. For instance,