Question

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

• A. All natural numbers $n>5$
• B. Only natural number $3$ $\le$ $n<15$
• C. All natural numbers $n$
• D. None

Answer 1

Answer: (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