^ADHATIC E(^ATI0N5. 3^5 



Then 5^T'^^{ . i by the question, 



X 



And izz ~— 3:^— V, Qr ArnrtTf-i from the second 



equation. 



Also ^f+i-f-jzir v+i Xj ^ron; the first equatjon. 

 Or 2jv-j-i::^;'^+j', 

 That is, ;;' — ■;.•=:; i. 

 Whence ;?* — y^-^zzi^ by completing the square 5 



Also y — y=:v/i|===v/^= — by evolution j 



Consequently ;t= [- y=z -*— •, 



A A I — v/5+3 

 And xzzz^'f-iz:^^ . 



7. There are four numbers in arithrnetical progression, 

 whereof the product of the two extremes is 45, and that 

 of the nieans 77; what are the numbers ? 



Let X = less extreme. 

 And y = common difference ; 

 Then x, x-^-y, X'{'2y, X-j-^y ^'^^^ be the four numbers, 



And 3 ^^^ +3;'=^ * +3^^=45 ? by the 



J I v^*- I 2 I > * \ question. 



ix+yXx-\;2yZ=x'+2>^y+V =11 J 



Whence 2y^ Z^'}'] — 451=332, and j)*:ryzzi6 by sub- 

 traction and division. 

 Or y=:\/ 1611=4 by evolution, 

 Therefore x'' '\- i^xyzzx"" ■^iix'iz.^^ by the first equation, 

 Also x^ -\-\2X-\-i^6z=.^'<^'\-2,6zz'6i bycompleting the square. 

 And x-^-bzzzs/ ^iz=z^]yy the extracting of roots. 



Consequently x-=.g-~^6:=zii 

 And the numbers are 3, 7, 11 and z.y 



8. To 



