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Question
1.2 + 2.3 + 3.4 + ... + n (n + 1) = \[\frac{n(n + 1)(n + 2)}{3}\]
Solution
Let P(n) be the given statement.
Now,
\[P(n) = 1 . 2 + 2 . 3 + 3 . 4 + . . . + n(n + 1) = \frac{n(n + 1)(n + 2)}{3}\]
\[\text{ Step } 1: \]
\[P(1) = 1 . 2 = 2 = \frac{1(1 + 1)(1 + 2)}{3}\]
\[\text{ Hence, P(1) is true } . \]
\[\text{ Step } 2: \]
\[\text{ Let P(m) be true } . \]
\[\text{ Then, } \]
\[1 . 2 + 2 . 3 + . . . + m(m + 1) = \frac{m(m + 1)(m + 2)}{3}\]
\[\text{ To prove: P(m + 1) is true .} \]
\[\text{ That is,} \]
\[1 . 2 + 2 . 3 + . . . + (m + 1)(m + 2) = \frac{(m + 1)(m + 2)(m + 3)}{3}\]
\[\text{ Now, P(m) is } \]
\[1 . 2 + 2 . 3 + . . . + m(m + 1) = \frac{m(m + 1)(m + 2)}{3}\]
\[ \Rightarrow 1 . 2 + 2 . 3 + . . . + m(m + 1) + (m + 1)(m + 2) = \frac{m(m + 1)(m + 2)}{3} + (m + 1)(m + 2)\]
\[ \Rightarrow P(m + 1) = \frac{(m + 1)(m + 2)(m + 3)}{3}\]
\[\text{ Thus, P(m + 1) is true .} \]
\[\text{ By the principle of mathematical induction, P(n) is true for all n } \in N . '\]
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