1/2*5+ 1/5*8 + 1/8*11 +.......................+
1/(3n-1)(3n+2)
= n/6n+4
prove by PMI
P(n) = n/(6n+4)
Prove it for (n = 1)
P(1) = 1/(6*1+4) = 1/10 = 1/(2 * 5)
Assume it for n = k
P(k) = k/(6k+4)
Now proving it for n = ko+1
P(k) + 1/{3(k+1)-1}{3(k+1)+2}
k/(6k+4)+1/{3k+3-1}{3k+3+2}
K/(6k+4)+1/{3k+2}{3k+5}
k/2(3k+2)+1/{3k+2}{3k+5}
1/(3k+2) [k/2 + 1/(3k+5)]
1/(3k+2) [k(3k+5) +2 / 2(3k+5)]
1/(3k+2) [(3k^2+5k+2) / (6k + 10)
1/(3k+2) [{3k^2+3k+2k+2} / {6k+10}
1/(3k+2) [{3k(k+1) + 2(k+1)} / (6k+10)
1/(3k+2) [(3k+2) (k+1) / (6k+10)
cancelled 3k+2
(k+1) / 6k+10
P (k+1) = k+1/6(k+1) + 4
=k+1/6k+10