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

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