Let,

\[P\text{ }\left( n \right)\text{ }=\text{ }1/1.2\text{ }+\text{ }1/2.3\text{ }+\text{ }1/3.4\text{ }+\text{ }\ldots \text{ }+\text{ }1/n\left( n+1 \right)\text{ }=\text{ }n/\left( n+1 \right)\]

For, \[n\text{ }=\text{ }1\]

\[P\text{ }\left( n \right)\text{ }=\text{ }1/1.2\text{ }=\text{ }1/1+1\]

\[1/2\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\]

P (n) is true for \[n\text{ }=\text{ }1\]

Let’s check for P (n) is true for \[n\text{ }=\text{ }k,\]

\[1/1.2\text{ }+\text{ }1/2.3\text{ }+\text{ }1/3.4\text{ }+\text{ }\ldots \text{ }+\text{ }1/k\left( k+1 \right)\text{ }+\text{ }k/\left( k+1 \right)\text{ }\left( k+2 \right)\text{ }=\text{ }\left( k+1 \right)/\left( k+2 \right)\]

Now,

\[1/1.2\text{ }+\text{ }1/2.3\text{ }+\text{ }1/3.4\text{ }+\text{ }\ldots \text{ }+\text{ }1/k\left( k+1 \right)\text{ }+\text{ }k/\left( k+1 \right)\text{ }\left( k+2 \right)\]

\[=\text{ }1/\left( k+1 \right)/\left( k+2 \right)\text{ }+\text{ }k/\left( k+1 \right)\]

\[=\text{ }1/\left( k+1 \right)\text{ }\left[ k\left( k+2 \right)+1 \right]/\left( k+2 \right)\]

\[=\text{ }1/\left( k+1 \right)\text{ }\left[ {{k}^{2}}~+\text{ }2k\text{ }+\text{ }1 \right]/\left( k+2 \right)\]

\[=1/\left( k+1 \right)\text{ }\left[ \left( k+1 \right)\text{ }\left( k+1 \right) \right]/\left( k+2 \right)\]

\[=\text{ }\left( k+1 \right)\text{ }/\text{ }\left( k+2 \right)\]

P (n) is true for \[n\text{ }=\text{ }k\text{ }+\text{ }1\]

Hence, P (n) is true for all n ∈ N.