SERIES AND MATHEMATICAL INDUCTION 5

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

The first statement, a; = 2 X 1 — 1, is evidently tru.

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

ar = 2k — 1
Now ar41 = ar + 2, as any odd number can be obtaind

from the preceding odd number by adding 2.
Hence by our assumption
a1 =QRek—1)+2=2k4+1=2(+1) -1
Thus again the kth statement in our series, ar = 2k — 1,
implies the & + 1th, ¢y = 2(k + 1) — 1.
Therfor the statements are all tru.
That is, the nth odd number starting with 1 is 2z — 1.

ARITHMETIC PROGRESSIONS

15. The series of even and of odd numbers just considerd
are particular cases of series called arithmetic series, or arith-
metic progressions.

Definition. An arithmetic progression may be defined as a
series of numbers of which the first is finite and any term of
which except the first is obtaind from the preceding term by
adding the same finite number to that preceding term. This
increment is of course the difference between any term except
the first and the preceding term and is on this account called
the common difference. The common difference is usually
denoted by d.

16. Thus, if ai, as, a3, -*+, @, +-+ is an arithmetic pro-
gression,
Ay = Q1 + d
a3 = Qs + d
ay=as + d
Gny1 = an + d

and @y —G1 =03 — QG2 =Q4— 3= *** =Qup1 — QG = -+ =0