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Question
n(n + 1) (n + 5) is a multiple of 3 for all n ∈ N.
Solution
Let P(n) be the given statement.
Now,
\[P(n): n(n + 1)(n + 5)\text{ is a multiple of } 3 . \]
\[\text{ Step1 } : \]
\[P(1): 1(1 + 1)(1 + 5) = 12 \]
\[\text{ It is a multiple of} 3 . \]
\[\text{ Hence, P(1) is true } . \]
\[\text{ Step2 } : \]
\[\text{ Let } P\left( m \right) \text{ be true . } \]
\[\text{ Then,} m\left( m + 1 \right)\left( m + 5 \right) \text{ is a multiple of } 3 . \]
\[\text{ Suppose} m\left( m + 1 \right)\left( m + 5 \right) = 3\lambda, \text{ where } \lambda \in N . \]
\[\text{ We have to show that } P\left( m + 1 \right) \text{ is true whenever P(m) is true } . \]
\[\text{ Now, } \]
\[P(m + 1) = \left( m + 1 \right)\left( m + 2 \right)\left( m + 6 \right)\]
\[ = m\left( m + 1 \right)\left( m + 6 \right) + 2\left( m + 1 \right)\left( m + 6 \right)\]
\[ = m\left( m + 1 \right)\left( m + 5 + 1 \right) + 2\left( m + 1 \right)\left( m + 6 \right)\]
\[ = m\left( m + 1 \right)\left( m + 5 \right) + m(m + 1) + 2\left( m + 1 \right)\left( m + 6 \right)\]
\[ = 3\lambda + \left( m + 1 \right)\left( m + 2m + 6 \right) \left[ From P(m) \right]\]
\[ = 3\lambda + 3\left( m + 1 \right)\left( m + 2 \right)\]
\[\text{ It is clearly a multiple of 3} . \]
\[\text{ Thus, P(m + 1) is true } . \]
\[\text{ By the principle of mathematical induction, P(n) is true for all n } \in N .\]
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