198 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Reversing the order of the terms in the second parenthesis,
d+d = (r — 1)(1-7’" 4+ 3 2y~ L 1)
k=2

=@ =142 142240 42224 2-741)
=(r—1)(1222---1)

Thus we have the theorem:

Th. Ifr,m,d,and d areasin § 482 and r > 2, then d + d’
is equal to (r — 1) times a number of m digits of which the first
and last are 1's and the others, if there are any, are 2’s.

484. The result obtaind in § 482, that

B R e T e =
> Aot 2 = == e I T e R = =

d+d = (r— 1)(1 + 2 fi: rk=1 4 1-7"),
k=2

DI

we can put in another form, which we will be able to descnbe
in words without using cases.
If » > 1, in which case we have the middle terms,

&

S b ot

T v

n n—1
since 3 Rt Rk
k=2

zzrk-l_zrk 1+Zrk

k=2

1 -‘
3
g
1 t
!
{
A
{
i
i

n n—1
Henced 4+ d' = (r — 1)(1 + X4 Tt ,n)
k=2 k=1

=(r—1) érk—l-l—érk)

O G