MATHEMATICAL RECREATIONS 199
= (r — 1)(2 po-tole g 3 r"—l)
=1 =1
=({r—-1)(r+1) > r?
k=1

= (2= 1) 3
k=1
If » = 1, in which case we do not have the middle terms,

d+d =@ —1)1 +7) =7

1=(2-1)3 r,
k=1

since, when n = 1, »* = 7 and

ol

n
YAl =3l = g0 = 1
k=1

k=1

Thus, whether » > 1 or n = 1,

d+d =r—-1)2Yrr1=(>(r—-1)3 m*
k=1 k=1
Hence we have the theorem:
Th. Ifr,m,d, and d’ areasin § 482, then d + d’ is equal to
7?2 — 1 times a number of m — 1 digits, all of which are I's.

485. As a consequence of the rules of §§ 480, 482 for @ — o’
and d + d’ we have the following rule:

Th. Given a number of m (two or more) digits written in a
notation whose radix 1is r, the first digit being greater than the
last; if
1st, we interchange the first and last digits,
2d, subtract the resulting number from the given number.
3d, interchange the first and last digits of the difference, regarding

it as also a number of m digits,
4th, add the resulting number to the difference;