SERIES AND MATHEMATICAL INDUCTION g

have

a =a, — d

a3=a,2—-d

a4=a3—d
andal—a2=a2—a3=a3—a4—_— e =0y — Ol = 0 =d
Also an=a — (n — 1)d

GEOMETRIC PROGRESSIONS

21. Def. A geometric progression may be defined as a
series of numbers of which the first is finite and non-zero and
any term of which except the first is obtaind from the pre-
ceding term by multiplying that preceding term by the same
finite non-zero number. This multiplier is of course the ratio
of any term except the first to the preceding term and is on
this account called the common ratio. The common ratio is
usually denoted by 7. :

For example, we have the geometric progressions

1,1, 1, <+ inwhich r=1:

1,2, 4, ---, mwhich £ = 2:

2, —2/3, 2]9, - <+, In which r == £k

1, — 1, — 4, -+, in which 7 = 4, the imaginary unit.

22. If a1, a3, a3, - -+, @y, - -+ is a geometric progression,

as = a7
a3 = aor
as = agr

and a: 101 =03:02 =04 :Q3 = *** =Qpy1 1A= +++ = ¢