114 
THE APPLICATION OF ALGEBRA 
n % . U + 1 
2/1 — 2 
24 
2 n l . n \ \ . n — 1 
12 
12 
Therefore 
n 1 . n 
— 1 . e l 
6 Y and e — 
—- ] is known 
12 
✓ 
u z / vir 1 
12«c— 12Ä* 
; whence x 
2 
** 1 i . e 
Example. Let the given number of terms be 6, their 
sum 33, and the sum of their squares 199; then, by 
writing these numbers, respectively, for //, b, and c, we 
shall have e — \ \ whence x — 2, and the required 
numbers 3, 4, 3, 6, 7, and 8. 
/ 
PROBLEM LXI II, 
Tiro post-boys A and B set out, at the same time, from 
two cities 500 miles asunder, in order to meet each other : 
A rides- 60 miles the first day, 55 the second, 50 the third, 
and so on, decreasing 5 miles every day : but B goes 40 
miles the first day, 45 the second, 50 the third. §c. in 
creasing 5 miles every day; now it is required to find in 
what number of days they will meet ? 
In order to have a general solution to this problem, 
let the first day’s distance of the post A be put = m, 
and the distance which he falls short each day of the 
preceding — d; also the first day’s distance of the post 
B - p, and the distance which he gains each day = e ; 
and let a: be the required number of days in which they 
meet: then the whole distance travelled by A will be 
expressed by the following arithmetical progression, 
m + m — d + m — 2d + m — 3d, &c. and that of B by 
p -f p -f e T p + ' 2e + p + 3e, &c. where each pro 
gression is to be continued to x terms. But the sum of 
the first of these progressions [by Sect. 10, Theor. 4.) iszr 
0 
x x x— 1 x e 
therefore these two last expressions, add- 
2