1.
`A= 1/(1.2.3) + 1/(2.3.4) + 1/(3.4.5)+...+1/(2014.2015.2016`
`A= 1/2(1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 +....+1/2014.2015 -1/2015.2016 )`
`A= 1/2 (1/1.2 - 1/2015.2016)`
`A= 1/2 ( 1/2 - 1/2015.2016)`
`A= 1/4 - 1/(2.2015.2016) < 1/4`
Vậy `A< 1/4`
2.
a) `x/3 + 1/y= 1/2`
`=> (xy)/(3y) + 3/(3y) = 1/2`
`=> (xy+3)/(3y) = 1/2`
`=> 2(xy+3) = 3y`
`=> 2xy + 6 = 3y`
`=> 2xy - 3y = -6`
`=> y(2x-3) = -6`
`=> y ; 2x - 3 in Ư(6)`
`=> y; 2x-3 in { 1;2;3;6 ; -1; -2;-3; -6}`
Vì `2x` chẵn `=> 2x-3` lẻ
`=> 2x-3 in { 1;3; -1; -3}`
+) `2x - 3 = 1; y = -6`
`=> x= 2 ; y =-6`
+) `2x -3 = -3 ; y= 2`
`=> x =0 ; y=2`
+) ` 2x -3 = 3 ; y = -2`
`=> x= 3; y= -2`
+) `2x - 3= -1 ; y= 6`
`=> x= 1; y= 6`
Vậy các cặp số `(x;y)` là : `(1;6) ; ( 3; -2) ; ( 2; -6) ; ( 0;2)`
b) `5xy + 2y +x =3`
`5(5xy + 2y +x) = 5.3`
`25xy + 10y + 5x = 15`
`5y(5x + 2) + 5x = 15`
`5y(5x +2) + (5x+2) -2 = 15`
`(5x+2)(5y+1) = 17`
`=>5x+2 ; 5y+1 in Ư(17)`
`=> 5x+2; 5y+1 in { 1;17; -1; -17}`
+) `5x + 2= 1; 5y+1 = 17`
`=> x= -1/5 ; y= 16/5`
+) `5x +2 = 17; 5y+1 = 1`
`=> x= 3; y= 0`
+) `5x+2 = -1; 5y+1 = -17`
`=> x= -3/5 ; y= -18/5`
+) `5x+2 = -17; 5y+1 = -1`
`=> x= -19/5 ; y= -2/5`
Vậy các cặp `(x;y)` thỏa mãn là : `(-19/5 ; -2/5)` ; `( -3/5; -18/5)` ; `( 3;0)` ; `( -1/5 ; 16/5)`
