Let,

\[P\text{ }\left( n \right)\text{ }=\text{ }1/3.5\text{ }

+\text{ }1/5.7\text{ }+\text{ }1/7.9\text{ }+\text{ }\ldots \text{ }

+\text{ }1/\left( 2n+1 \right)\left( 2n+3 \right)\text{ }=\text{ }n/3\left( 2n+3 \right)\]

Let us check for \[n\text{ }=\text{ }1,\]

\[P\text{ }\left( 1 \right):\text{ }1/3.5\text{ }=\text{ }1/3\left( 2.1+3 \right)\]

\[:\text{ }1/15\text{ }=\text{ }1/15\]

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

Now, let us check for P (n) is true for\[n\text{ }=\text{ }k\] , and have to prove that \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]

is true.

\[P\text{ }\left( k \right)\text{ }=\text{ }1/3.5\text{ }+\text{ }1/5.7\text{ }+\text{ }1/7.9\text{ }

+\text{ }\ldots \text{ }+\text{ }1/\left( 2k+1 \right)\left( 2k+3 \right)\text{ }=\text{ }k/3\left

( 2k+3 \right)\text{ }\ldots \text{ }\left( i \right)\]

So,

\[1/3.5\text{ }+\text{ }1/5.7\text{ }+\text{ }1/7.9\text{ }+\text{ }\ldots \text{ }

+\text{ }1/\left( 2k+1 \right)\left( 2k+3 \right)\text{ }+\text{ }1/\left[ 2\left( k+1 \right)+1 \right]\left[ 2\left( k+1 \right)+3 \right]\]

\[1/3.5\text{ }+\text{ }1/5.7\text{ }+\text{ }1/7.9\text{ }+\text{ }\ldots \text{ }

+\text{ }1/\left( 2k+1 \right)\left( 2k+3 \right)\text{ }+\text{ }1/\left( 2k+3 \right)\left( 2k+5 \right)\]

Now substituting the value of P (k) we get,

\[=\text{ }k/3\left( 2k+3 \right)\text{ }+\text{ }1/\left( 2k+3 \right)\left( 2k+5 \right)\]

\[=\text{ }\left[ k\left( 2k+5 \right)+3 \right]\text{ }/\text{ }\left[ 3\left( 2k+3 \right)\left( 2k+5 \right) \right]\]

\[=\text{ }\left( k+1 \right)\text{ }/\text{ }\left[ 3\left( 2\left( k+1 \right)+3 \right) \right]\]

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

Hence, P (n) is true for all n ∈ N.