n (n+1) (n+5) is a multiple of 3 is true for - Vidyadip

MCQ

$n (n+1) (n+5)$ is a multiple of $3$ is true for

A
All natural numbers $n > 5$
B
Only natural number $3 ≤ n < 15$
✓
All natural numbers $n$
D
None

✓

Answer

Correct option: C.

All natural numbers $n$

Let the statement be denoted by $p(n)$
i.e., $P(n) : n (n+1) (n+5)$ is a multiple of $3$
For $n = 1, n(n+1) (n+5) = 1.2.6 = 12 = 3.4$
$P(n)$ is true for $n = 1$
Suppose $p(k)$ is true for $n = k$
i.e. $k(k+1) (k+5) =3m ($let$)$ or $k^3 + 6k^2 + 5k = 3m ... (i)$
Replacing $k$ by $k+1,$ we get
$(k+1) (k+2) (k+6) = k (k^2+ 8k + 12) + (k^2+ 8k + 12) $
$k^3+ 9k^2+ 20k + 12 = (k^3+ 6k^2+ 5k) + (3k^2+15k+12)$
$= 3m + 3k^2+ 15k + 12 [$from $(i)]$
$= 3 (m + k^2+ 5k + 4)$
i.e. $(k+1) (k+2) (k+6)$ is a multiple of $3$
i.e. $P(k+1)$ is multiple of $3,$ if $P(k)$ is a multiple of $3$
i.e. $P(k+1)$ is true whenever $P(k)$ is true.
Hence $P(n)$ is true for all $n \in N$

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