State whether following statement is true or false.
By using the principle of mathematical induction we can proove that $n (n + 1) (n + 5)$ is a multiple of $3$ .

True

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False

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Solution

The correct option is A True
Let $p (n) = n (n + 1) (n + 5) = 3 d$
where $d ϵ N$
For $n = 1$

$L H S = 1 (1 + 1) (1 + 5)$
$= 1 (2) (6)$
$= 12$

$R H S$ $= 3 d = 3 \times 4 = 12$

$∴ p (n)$ is true for $n = 1$

$∴ n (n + 1) (n + 5)$ is a multiple of $3$ .

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Similar questions

Prove the following by using the principle of mathematical induction for all n ∈ N: n (n + 1) (n + 5) is a multiple of 3.

Using the principle of mathematical induction, prove that

$1.2.3 + 2.3.4 + 3.4.5 + . . . + n (n + 1) (n + 2) = \frac{n (n + 1) (n + 2) (n + 3)}{4},$ for all $n \in N .$

n \in N : 1.2.3 + 2.3.4 + . . . . . . + n (n + 1) (n + 2) = \frac{n (n + 1) (n + 2) (n + 3)}{4}

n \in N

1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \dots + n (n + 1) (n + 2)

= \frac{n (n + 1) (n + 2) (n + 3)}{4}

n \in N :

1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots + n (n + 1) = \frac{n (n + 1) (n + 2)}{3}

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