BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 95 
Here we observe that the Ist, 7th, and 10th, have the same sum; so also 
have 2d, 6th, and 11th, and so on. 
But the same series expressed in the tredecimal notation, and continued to 
13 terms, is ; 
1, 3, 6, 1,, 12, 18, 22, 21,, 36, 43, 51, 60, 70 
Sums, 1, 35 6, Le 3, 9, 4, Le 9, ac 6, 6, 7 
Here the 13th term has a sum, 7, different from the first. 
But if we take p=2, n»—1=12, and s=11, 
then pa—1—s+1=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 Ist, 7th, 
and 10th terms have one for their ultimate sum, but, also the 4th, 13th, &c. 
This happens, because in the decimal scale, fee 



ae 
but the “= * " term =— = ="? nate? Us and, consequently, its ultimate 
sum is 1. 
Prop. XV. 
If the general term of any series be az"+ba"—14ee"—*? . . . . JC; then 
evidently, if x+n2—1 be substituted for 2, the result will be the original term + a 
multiple of »—1. Or, as in all the preceding forms, the same ultimate sum will 
recur after »—1 terms. 
If the general term be quadratic = az?+b2+c. 
Let «,=y—z, then az*=ay?—2ary+ax? 
oa by—bz 
Os c 
Therefore ax +ba,+c=ax2+b2+ c—2ayt+2batay+by 
=ax?+bat+c+ayt+bxy—2e. 
Now, let y be assumed such, that ay+=p.—1, then the 2 and a," terms 
will have same ultimate sum. 
Ex. Let 22 +382+1 be the general term. Substitute for x successively 0, 1, 2, 
&c., we have the series, 
1, 5, TU, 195 29, 41, 55,71, 89, 109, &e. 
Ultimate Sums, bh DO ee be Se Bod ke: 
Here a=1,5=3. If, therefore, y+8=9, or y=6, the two terms in which the 
VOL. XVI. PART II. 2B 
