`a)`

Ta có: `1/(sqrt(k)+sqrt(k+1)) = (sqrt(k+1)-sqrt(k))/((k+1)-k) = sqrt(k+1)-sqrt(k)`

`A = (sqrt(2)-sqrt(1))+(sqrt(3)-sqrt(2))+...+(sqrt(n+1)-sqrt(n))`

`A = -sqrt(1) + sqrt(n+1)`

`A = sqrt(n+1) - 1`

``

`b)`

Ta có: `1/(sqrt(k)+sqrt(k+2)) = (sqrt(k+2)-sqrt(k))/((k+2)-k) = (sqrt(k+2)-sqrt(k))/2`

`B = 1/2 * [(sqrt(3)-sqrt(1)) + (sqrt(4)-sqrt(2)) + (sqrt(5)-sqrt(3)) + ... + (sqrt(n+2)-sqrt(n))]`

`B = 1/2 * [-sqrt(1) - sqrt(2) + sqrt(n+1) + sqrt(n+2)]`

`B = (sqrt(n+1) + sqrt(n+2) - 1 - sqrt(2))/2`

``

`c)`

Ta có: `1/(sqrt(k)-sqrt(k+1)) = (sqrt(k)+sqrt(k+1))/(k-(k+1)) = -(sqrt(k)+sqrt(k+1))`

`C = -(sqrt(1)+sqrt(2)) - (-(sqrt(2)+sqrt(3))) + (-(sqrt(3)+sqrt(4))) - ... + (-(sqrt(2n+1)+sqrt(2n+2)))`

`C = -sqrt(1)-sqrt(2) + sqrt(2)+sqrt(3) - sqrt(3)-sqrt(4) + ... - sqrt(2n+1)-sqrt(2n+2)`

`C = -sqrt(1) - sqrt(2n+2)`

`C = -1 - sqrt(2n+2)`

``

`d)`

Ta có: `1/((k+1)sqrt(k)+ksqrt(k+1)) = 1/(sqrt(k)sqrt(k+1)(sqrt(k+1)+sqrt(k)))`

`=1/((k+1)sqrt(k)+ksqrt(k+1)) = ((k+1)sqrt(k)-ksqrt(k+1))/(((k+1)sqrt(k))^2-(ksqrt(k+1))^2)`

`= ((k+1)sqrt(k)-ksqrt(k+1))/(k(k+1)^2-k^2(k+1)) = ((k+1)sqrt(k)-ksqrt(k+1))/(k(k+1))`

`= (k+1)sqrt(k)/(k(k+1)) - ksqrt(k+1)/(k(k+1)) = 1/sqrt(k) - 1/sqrt(k+1)`

`D = (1/sqrt(1)-1/sqrt(2)) + (1/sqrt(2)-1/sqrt(3)) + ... + (1/sqrt(n)-1/sqrt(n+1))`

`D = 1/sqrt(1) - 1/sqrt(n+1)`

`D = 1 - 1/sqrt(n+1)`