SPECIAL TOPICS IN THEORETICAL ARITHMETIC

23. Evidently a geometric progression is uniquely determind
when its first term and common ratio are given.

Th. If a is the first term, it can be easily proved by mathe-
matical induction that the nth term is ar .

24. Of course a geometric progression could just as well be
defined in terms of division instead of multiplication.

If the common divisor is then denoted by 7, this 7 is the
reciprocal of the 7 of § 21 and

Qs = Q1 . 7
as = az . 7
ag = Qg . r

a1 02 = Q3 A3 — A3 : Q4 = *°*° = Qn :Qpp1 = 7

and e =2 vt

HARMONIC PROGRESSIONS

25. Def. A harmonic progression is a series of numbers
whose reciprocals are an arithmetic progression.

For example, since the natural numbers 1, 2, 3, --- form
an arithmetic progression, their reciprocals 1, 1/2, 1/3,
form a harmonic progression.

Similarly a series, all of whose terms are 3’s, is a harmonic
progression.

26. We have seen that the nth terms of some series can be
exprest in terms of # by means of well known operations.

Conversely, if the nth term is exprest in terms of # by
well known operations, the whole series is given.

For example, if a, = n(n + 1)/2,

s =1, az =3, as = 6, as = 10, etc.