for the P(k+1) you must have:

1/2(5) +1/5(8)+.......................+1/(3n-1)(3n+2) + 1/(3n+2)(3n+5) =

=(n+1)/6[(n+1)+4]= (n+1)/2(3n+5)........................................... ............................1

IF (1) is satisfied then the P(k+1) step is satisfied.

Since you already have : 1/2(5) +....................+1/(3n-1)(3n+2)= n/(6n+4) ,substitute that into (1) and you get:

n/(6n+4) + 1/(3n+2)(3n+5) = (n+1)/2(3n+5) or:

n/2(3n+2) + 1/(3n+2)(3n+5) = (n+1)/2(3n+5).

DO THE calculations and you will see that the equation is satisfied