$1.2+2.3+3.4+\ldots+n(n+1)=\frac{n(n+1)(n+2)}{3}$
Let P(n) be the given statement.
Now,
$P(n)=1.2+2.3+3.4+\ldots+n(n+1)=\frac{n(n+1)(n+2)}{3}$
Step 1:
$P(1)=1.2=2=\frac{1(1+1)(1+2)}{3}$
Hence, $P(1)$ is true.
Step 2:
Let $P(m)$ be true.
Then,
$1.2+2.3+\ldots+m(m+1)=\frac{m(m+1)(m+2)}{3}$
To prove: $P(m+1)$ is true.
That is,
$1.2+2.3+\ldots+(m+1)(m+2)=\frac{(m+1)(m+2)(m+3)}{3}$
Now, $P(m)$ is
$1.2+2.3+\ldots+m(m+1)=\frac{m(m+1)(m+2)}{3}$
$\Rightarrow 1.2+2.3+\ldots+m(m+1)+(m+1)(m+2)=\frac{m(m+1)(m+2)}{3}+(m+1)(m+2)$
$\Rightarrow P(m+1)=\frac{(m+1)(m+2)(m+3)}{3}$
Thus, $P(m+1)$ is true.
$B y$ the $p$ rinciple of $m$ athematical induction, $P(n)$ is true for all $n \in N$.
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