N(N + 1) (N + 5) is a Multiple of 3 for All N ∈ N.

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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Principle of Mathematical Induction

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Q 24 Q 23 Q 25

Chapter 12: Mathematical Induction - Exercise 12.2 [Page 28]

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R.D. Sharma Mathematics [English] Class 11

Chapter 12 Mathematical Induction
Exercise 12.2 | Q 24 | Page 28

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