Question

n(n+1) (n+5) is a multiple of 3​

Answer 1

Step-by-step explanation:

We will prove it by using the formula of mathematical induction for all n ϵ N

Let P(n)=n(n+1)(n+5)=3d where d ϵ N

For n=1

P(1)=1(2)(6)=12 which is divisible by 3

Let P(k) is true

P(k)=k(k+1)(k+5)=3m where m ϵ N

⟹k

3

+6k

2

+5k=3m

⟹k

3

=−6k

2

−5k+3m

Now we will prove that P(k+1) is true

P(k+1)=(k+1)(k+2)(k+6)=k

3

+9k

2

+20k+12

Putting the value of k

3

in above equation we get,

(3m−6k

2

−5k)+9k

2

+20k+12

=3m+3k

2

+15k+12

=3(m+k

2

+5k+4)

3r where r=m+k

2

+5k+4

Since P(k+1) is true whenever P(k) is true.

So, by the principle of induction, P(n) is divisible by 3 for all n ϵ N