Question

Q27. Using PMI, show that n (n + 1) (n + 5) is divisible by 3.​

Answer 1

Answer:

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

⟹k3+6k2+5k=3m

⟹k3= −6k2−5k+3m

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

P(k+1)=(k+1)(k+2)(k+6)=k3+9k2+20k+12

Putting the value of k3

in above equation we get,

(3m−6k2−5k)+9k2+20k+12

=3m+3k2+15k+12

=3(m+k2+5k+4)

3r where r=m+k2+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

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