5 



\66 ALGEBRA. 



8. To find three numbers in geometrical progression, 

 whose sum shall be 14, and the sum of their squares 84. 



Let X, y and z be the numbers sought ; 

 Then xzzny^ by the nature of proportion, 



A"d [itlP^'ts^] by the question, 



But x-{-zzzi4 — y by the second equation, 



Aiid x"" '■^2Xz-{-z^ zi:ig6-^2oy-j'y^ by squaring both sides, 



Or X^-{-z^"{'2y'' = 196 — 28j-f-;'* by putting 2y' for its 



equal 2^2 ; 



That is, X * -\-z ' -{ry * = 1 96— ^28^ by subtraction. 



Or 196 — 28j'=o4rby equality ; 



196 — 84 

 Hence ^'=x ;:; — —14 by transposition and division. 



Again, Afzzrv ' i:z 1 6, or ;v— — by the first equation, 



z 



And x-\~j-\'Z=: [-4-J-2:= 14 by the 2d equation. 



Or i6-[-42:'4"2:'' == 14-J or s^ — -icz = — 16., 

 Whence z^ — ioz-{-25=;: 25 — 16=9 by completing the 



square ; 

 Ar>d 2—5 = ^/9 = 3, or% = 3-f5=r8i 

 Consequently ;v= 14 — y — 2=14 — 4 — 8=2, and the 

 Ti umbers are 2, 4, 8. 



9. The sum .(j-) and the product (/>) of any two num- 

 bers being given j to find the sum of the squares, cubes, 

 biquadrates, &c. of those numbers. 



Let the tviro numbers be denoted by x and y ; 



Then will j '^"^•^^^]- by the question, 



' ^ 



But x-^-yl z:rx^-\~2xy-\~y^z=s^ by involution^ 



And a;* -|* 2^^+^' — 2,tj— j* — 2/> by subtraction. 



That is, x^-{-y'^zzs'^ — zpzusnm of the squares. 



Again 



