Bài 19:
1) `-9/x = x^2/3`
`=> -9.3 = x.x^2`
`=> -27 = x^3`
`=> (-3)^3 = x^3`
`=> x= -3`
Vậy `x = -3`
2) `y/-25= 1/-y`
`=> y.(-y) = -25`
`=> -y^2= -25`
`=> y^2= 25`
`=>` \(\left[ \begin{array}{l}y=5\\x=-5\end{array} \right.\)
Vậy `y=5` hoặc `y= -5`
3) `3/x + y/3= 5/6`
`=> 9/(3x)+ (xy)/(3x) = 5/6`
`=> (9+xy)/(3x) = 5/6`
`=> 6( 9+xy) = 5. 3x`
`=> 54 + 6xy = 15x`
`=> 15x - 6xy = 54`
`=> 3x( 5 - 2y) = 54`
`=> 3x ; 5-2y in Ư(54) ={± 1;±2; ±3;± 9; ±27; ±6; ±18; ±54 }`
Mà `5-2y` là số lẻ `=> 5 -2y in { ±1; ±3 ; ±9; ±27}`
+) Nếu `5- 2y =1 ; 3x = 54`
`=> 2y = 4 ; x= 18`
`=> y= 2 ; x= 18`
+) Nếu `5-2y= -1 ; 3x = -54`
`=> 2y= 6 ; x= -54 :3`
`=> y= 3; x= -18`
+) Nếu `5- 2y = 3 ; 3x = 18`
`=> 2y= 2 ; x= 6`
`=> y= 1 ; x= 6`
+) Nếu `5- 2y =-3 ; 3x= -18`
`=> 2y= 8 ; x= -18:3`
`=> y= 4 ; x= -6`
+) Nếu `5-2y= 9 ; 3x = 6`
`=> 2y = -4 ; x= 6:3`
`=> y= -2; x= 2`
+) Nếu `5-2y = -9 ; 3x= -6`
`=> 2y = 14 ; x= -6:3`
`=> y= 7; x= -2`
+)Nếu `5-2y= 27 ; 3x = 2`
`=> 2y = -22 ; x = 2 /3`
`=> y= -11 ; x = 2/3` ( loại do `x ∉Z`)
+) Nếu `5- 2y= -27 ; 3x= -2`
`=> 2y= 32 ; x= -2/3`
`=> y= 16; x= -2/3` (loại do `x ∉Z`)
Vậy các cặp số `(x;y)` thỏa mãn là: `(-2; 7) ;(-6;4) ;( 6;1) ;(-18 ;3 ); (18;2); (-2;2)`
Bài 15:
a) `S_1 = 5/1.4 + 5/4.7 +...+ 5/97.100`
`S_1 : 5 = 1/1.4 + 1/4.7 +...+ 1/97.100`
`S_1 : 5. 3 = 3/1.4 + 3/4.7 +...+ 3/97.100`
`S_1 : 5 .3 = 1 - 1/4 + 1/4 -1/7 +...+ 1/97-1/100`
`S_1 : 5 .3 = 1- 1/100`
`S_1 = 99/100 : 3 .5`
`S_1= 33/20`
Vậy `S_1 =33/20`
b) `S_2 = 1/15 + 1/35 + 1/63+...+1/2499`
`S_2 = 1/3.5+ 1/5.7 + 1/7.9 +...+ 1/49.51`
`2 S_2 = 2/3.5 + 2/5.7 +...+ 2/49.51`
`2S_2 = 1/3-1/5+1/5-1/7 +...+ 1/49 - 1/51`
`2S_2= 1/3 - 1/51`
`2 S_2= 16/51`
`S_2= 16/51 :2`
`S_2= 16/51 . 1/2`
`S_2= 8/51`
Vậy `S_2= 8/51`
c) `S_3 = 1/14 + 1/35+...+1/350`
`S_3= 2/28 + 2/70 +...+ 2/700`
`S_3 = 2( 1/28 + 1/70 + ...+ 1/700)`
`S_3 = 2( 1/4.7 + 1/7.10 +....+ /25.28)`
`S_3 .3 = 2( 3/4.7 + 3/7.10+...+ 3/25.28)`
`S_3 .3 = 2( 1/4 -1/7 + 1/7 - 1/10+...+ 1/25-1/28)`
`S_3 .3 = 2 ( 1/4 - 1/28)`
`S_3 .3= 3/7`
`S_3= 3/7 :3`
`S_3 = 3/7 .1/3`
`S_3= 1/7`
Vậy `S_3 =1/7`
d) `S_4 = 1+ 1/2+ 1/4 + 1/8 +...+ 1/1024`
`S_4 = 1+ 1/2 + 1/2^2 + 1/2^3 +...+ 1/2^10`
`1/2 S_4 = 1/2 + 1/2^2 + 1/2^3 +...+1/2^11`
`S_4 - 1/2 S_4 =1 + 1/2 + 1/2^2 +...+ 1/2^10 - 1/2 - 1/2^2-...-1/2^11`
`1/2 S_4 = 1 - 1/2^11`
`S_4= (1- 1/2^11) : 1/2`
`S_4= 2- 1/2^10`
Vậy `S_4= 2- 1/2^10`
2) a) `x- 20/11.13 - 20/13.15 -...-20/53.55 = 3/11`
`x- (20/11.13 + 20/13.15 +....+ 20/53.55) = 3/11`
`x - 10(2/11.13 + 2/13.15 +...+ 1/53.55) = 3/11`
`x- 10( 1/11 - 1/13 + 1/13 -...+ 1/53 - 1/55)= 3/11`
`x - 10( 1/11 -1/55) = 3/11`
`x - 8/11 = 3/11`
`x = 3/11 + 8/11`
`x =1`
Vậy `x= 1`
b) `1/21 + 1/28+...+ 2/(x(x+1)) =2/9`
`1/2( 1/21 + 1/28 +....+ 2/(x(x+1)) = 2/9. 1/2`
`1/42 + 1/56+...+ 1/(x(x+1)) = 1/9`
`1/6.7 + 1/7.8 +...+ 1/(x(x+1))= 1/9`
`1/6 - 1/7 + 1/7 +...+ 1/x - 1/(x+1) = 1/9`
`1/6 - 1/(x+1) = 1/9`
`1/(x+1) = 1/6- 1/9 =1/18`
`=> x + 1 =18`
`=> x = 17`
Vậy `x= 17`
