BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 95 



Here we observe that the 1st, 7th, and 10th, have the same sum ; so also 

 have 2d, 6th, and Ilth, and so on. 



But the same series expressed in the tredecimal notation, and continued to 

 13 terms, is 



1, 3, 6, lo, 12, 18, 22, 21„, 36, 43, 51, 60, 70 



Sums, 1, 3, 6, lo, 3, 9, 4, 1„ 9, 7, 6, 6, 7 



Here the 13th term has a sum, 7, different from the iirst. 



But if we take jt)=2, n—l=12, and «=11, 

 thenjo.M-l-sTI=24-12=12; therefore 11th and 12th have same sum. 



If s=10, 26-11=13, therefore 10th and 13th have same sum, and so on. 



Note. — In the decimal series it may be observed, that not only the 1st, 7th, 

 and 10th terms have one for then- ultimate sum, but also the 4th, 13th, &c. 



This happens, because in the decimal scale, i—'^—- ; 



w— 2 n 

 but the ^^^ term = — = — — ^~% ^ = ^''^~ ■ =n, and, consequently, its ultimate 

 sum is 1. 



Prop. XV. 



If the general term of any series be a «" + i a;"' - ' + c «"' - = . . . . / ; then 

 evidently, if a;+re-l be substituted for x, the result will be the original term + a 

 multiple of M— 1- Or, as in aU the preceding forms, the same ultimate sum will 

 recur after n-1 terms. 



If the general term be quadratic = az- + bz + c. 



Tijet x^=i/—x, then axl = at/^ — 2 axy + aa;' 

 bcc^-= by — bx 



c = c 



Therefore ax\-irbxi+c=ax'' + bx+ c—2ay + 2b.z + ay + b.y 



=az'' +bx+c + a}/ + b xy— 2 a;. 



Now, let 7/ be assumed such, that ai/ + b=p.n-l, then the a;^ and Xi^^ terms 

 wiU have same ultimate smn. 



Ex. Let 2:= + 3 a; + 1 be the general term. Substitute for x successively 0, 1, 2, 

 &c., we have the series, 



1, 5, 11, 19, 29. 41, 55, 71, 89, 109, &c. 

 Ultimate smns, l, 5, 2, 1, 2, 5, l, 8, 8, 1, &c. 



Here a=l, 6=3. If, therefore, y + 3=9. or ^=6, the two terms in which the 



VOL. XVI. PAET II. 2 B 



