1/1×2+1/2×3+1/3×4....... 1/n(n+1)=n/n+1

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1/(1x2) + 1/(2x3) + 1/(3x4) + ...+ 1/(n(n+1) = n/(n+1)
n =1
1/(1(1+1)) = 1/(1+1)
1/2 = 1/2

n = k
1/(1x2) + 1/(2x3) + 1/(3x4) + ...+ 1/ k(k+1) = k/ (k+1)

n = k +1
1/(1x2) + 1/(2x3) + 1/ (3x4) + ...+ 1/ k(k+1) + 1/ { (k+1)(k+1+1)} = (k+1)/(k+1+1)
k/(k+1) + 1/ (k+1)(k+2)  = (k+1) / (k+2)
(k(k+2) + 1 )/(k+1)(k+2)  = (k+1) / (k+2)
(k² + 2k +1)  / (k+1)(k+2)  = (k+1) / (k+2)
(k+1)(k+1) / (k+1)(k+2)  = (k+1) / (k+2)
(k+1)/ (k+2) = (k+1) / (k +2)