116 
THE APPLICATION OF ALGEBRA 
— 
Let (1) the common difference of the progression 8, 
12, 16, &c. be put — c, and the first term thereof minus 
the said common difference — m, and let the number of 
terms, or the days each person travels, be expressed by 
x : then the sum of that progression, or the number ot 
miles which A travels will be x x m I - X X ^ 1 --— 
(by Sect. 10, Tlteor. 4.). And (by what follows hereafter) 
the sum of the progression 1+4 f 9 .... x 1 , or the dis 
tance travelled by B, will appear to be 
x / x I l / 2 r i l 
therefore, by the question, we have 
X / r fl X i 1 
— jnx + * y X 1 1 which, divided by x and con- 
. . 2x* + 3x + 1 ex f- e , 
tracted, gives ^ — m 1 — ; whence 
3.x 3ex , 3c 1 j , 
- : and, by com- 
a o’ ’ J 
3x 3ex 
~2 2 
* +-r 
2 
pleting the square, x 5 
9e 2 
16 1 
, 3e 
3m + — 
48m + 1 
+ 6e + 9C 1 
16 
* + T 
. / ' 
3e _ 
~4~ ~ 
9 
18e 
16 
“"T6* + 
18e 
, 9e 1 
~i6 
+ ‘iT ~ 
16 
; whence 
-, and x — 
== 7, the number of 
days required. 
PROBLEM LX V. 
The sum of the squares ( a J, and the continual product 
fbj, of four numbers ifi arithmetical progression being 
given; to find the numbers.