TO THE RESOLUTION OF PROBLEMS. 
121 
v 2 4 aa 
else is readily found. 
✓ 
: whence every thing 
PROBLEM LXXI. 
The sum (a) and the sum of the squares (b) office 
numbers, in geometrical progression, being given; to find 
the numbers, 
Let the three middle numbers be denoted by x, y, 
and 2: then the two extreme ones will be and —; 
y V * 
and therefore we shall have 
xx 
( , 
y 
by the question. 
V 
Put x + z — u ; then, by theilrst equation, —- 
y y 
— a — u — y. Wherefore, seeing the sum of the two 
extremes are expressed by a — u—y, and their rectangle 
by y 1 (see Theor. 7. Sect. 10), the sum of their squares 
will {by Prob. 68) be — a — u — y P — 2if : and, in the 
very same manner, the sum of the squares of the two 
terms {x and 2) adjacent to the middle one [y] will be 
- u 1 — 2if. Whence, by substituting these values, 
our equations become U ~ !l + u f y — a, and 
a — u — y p — <2if + u 2 — <2y 2 + y 2 — b ; which, by 
^eduction are changed to 1 
aa — 2 an — 2 ay + 2 uu + 2 uy — 2 yy — b, 
and ay — uu — uy f yy — 0. 
To the former of which add the double of the latter, 
^o shall aa — 2au — b; and therefore u zz — — 
2 2a