SERIES AND MATHEMATICAL INDUCTION 13

. We can similarly show that

12423434+ - +nn+1) =nn+ 1)(n+2)/3
222 23 ey gl )
1 1 1 1 n
1223732 " +n(n+1) S
Also we can show that the sum of the first n terms of an arith-
metic progression whose first term is a and common difference d
is (n/2)(2a 4+ (n — 1)d); and that, in a geometric progression
whose first term is a, if the common ratio is unity, the sum of
the first n terms is na(*), but if the common ratio r is not unity,
the sum s
a(l — rm)
L=y

35. It may here be remarkt that because a formula gives
the sum of the first # terms of a series for several values of #,
it does not at all follow that this formula will give the sum
for all values.

For example, if S, is the sum of the first # terms of the
series

I W RO o~ 1

Sn=2n*—=3n+2, if n=1 or n =2, but not if n = 3.
Also S, =n® — 5n2+ 11n — 6, if n = 1, 2, or 3, but not if
n = 4.

It would be easy to find a function of #» which would equal
Sa form =1, 2, 3, or 4, but not for n = 5.

However .S, is equal to #? for all values of 7, as we have
seen.

36. This is only another illustration of the fact that two
series may agree in any finite number of terms at the begin-
ning and still not be the same series.

(1) This case seems to be disregarded by authors generally.