Question

$n(n+1)(n+5)$ is a multiple of 3 for all $nϵN$.

Answer 1

Let P(n) : $n(n+1)(n+5)$ is a multiple of 3 for all $nϵN$.

For n = 1

1.(1 + 1) (1 + 5)

$=\left(2\right)\left(6\right)$\

$=12$

It is a multiple of 3

Let P(n) is true for n = k

k(k + 1)(k + 5) is a multiple of 3

k(k + 1)(k + 5) = 3 $\lambda$

We have to show that.

(k + 1)[(k + 1) + 1][(k + 1) + 5] is a multiple of 3

(k + 1)[(k + 1) + 1] [(k + 1) + 5] = $3\mu$

Now,

(k + 1) (k + 2) [(k + 1) + 5]

$=\left[k\right(k+1)+2(k+1\left)\right]\left[\right(k+5)+1]$

$=k(k+1)(k+5)+k(k+1)+2(k+1)(k+5)+2(k+1)$

$=3\lambda +k^2+k+2(k^2+6k+5)+2k+2$

[Using equation (1)]

$=3\lambda +k^2+k+2k^2+12k+10+2k+2$

$=6\lambda +3k^2+15k+12$

$=3(\lambda +k^2+5k+4)$

$=3\mu$

$\Rightarrow$ P(n) is true for n = k + 1

$\Rightarrow$ P(n) is true for all $nϵN$ by PMI