Question -
Answer -
Let P (n) = 1/1.2 + 1/2.3 + 1/3.4 + тАж + 1/n(n+1) = n/(n+1)
For, n = 1
P (n) = 1/1.2 = 1/1+1
1/2 = 1/2
P (n) is true for n = 1
LetтАЩs check for P (n) is true for n = k,
1/1.2 + 1/2.3 + 1/3.4 + тАж + 1/k(k+1) + k/(k+1) (k+2) =(k+1)/(k+2)
Then,
1/1.2 + 1/2.3 + 1/3.4 + тАж + 1/k(k+1) + k/(k+1) (k+2)
= 1/(k+1)/(k+2) + k/(k+1)
= 1/(k+1) [k(k+2)+1]/(k+2)
= 1/(k+1) [k2┬а+ 2k + 1]/(k+2)
=1/(k+1) [(k+1) (k+1)]/(k+2)
= (k+1) / (k+2)
P (n) is true for n = k + 1
Hence, P (n) is true for all n тИИ N.