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6 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

17. The series of even numbers 2, 4, 6, - -+ is an arithmetic
progression whose first term is 2 and whose common difference
is 2: the series of odd numbers 1, 3, 5, - - is an arithmetic

progression whose first term is 1 and common difference 2.

18. It is evident that, if the first term of an arithmetic
progression and the common difference are given, the arith-
metic progression is uniquely determind.

Th. In fact, if a s the first term, the nih term, Qn, 1S
a+ (n— 1)d

To prove this statement is equivalent to proving that the
series of statements
a=a+ Q1 —1)d
a =0+ (2 — 1)d
az = a + (3 = 1)d

|

an—;‘-a+(n—1)d

are all tru.
Evidently the first is tru.
Moreover
ak=a+(k—1)d:>
gl =g &= [a + (B — 1)d] + d
—a+[(k+1) —11d
So the kth statement, ar = @ + (k — 1)d
implies the k + 1th, @r1 =@ +[(k+1)—1]d
Therfor the statements are all tru.
That is, a» = a + (7 — 1)d.
19. An arithmetic progression might just as well be defined
as a series of numbers of which the first is finite and any term

of which after the first is obtaind from the preceding term
by subtracting the same finite number from that preceding

term.

20. If we denote this common decrement by d, then d
stands for the opposit of what it stood for in § 15 and we