Question

$\huge\underline\mathcal\red{Question}$
$prove \: by \: mathematical \: \\ induction \: that \\ \\ n(n + 1)(n + 5) \: is \: a \: \\ multiple \: of \: 3$​

Answer 1

Question:

Prove by Mathematical induction that n(n+1)(n+5) is a multiple of 3.

Theory :

Statements involving mathematical relations are known as the Mathematical statement.

Solution :

Let p(n) be the statement that n(n+1)(n+5) is multiple of 3.

$\sf\:p(n)=n(n+1)(n+5)$ is multiple of 3

Step 1

Prove that the statement p(1) is true , thus

$\sf\:p(n)=n(n+1)(n+5)$

$\sf\implies\:p(1)=1(1+1)(1+5)$

$\sf\implies\:p(1)=6$ which is multiple of 3

Therefore , p(1) is true

Step 2

Assume that the statement p(m) is true , thus

Let p(m) be true . Then ,

$\sf\:m(m+1)(m+5)$ is a multiple of 3

$\sf\:m(m+1)(m+5)=3\lambda,\:for\:\lambda\in\:N$

$\sf\:m(m+1)(m+5)=3\lambda$

$\sf\:(m^2+m)(m+5)=3\lambda$

$\sf\:m^3+6m^2+5m=3\lambda$

$\sf\:m^3=3\lambda-6m^2-5m..(1)$

Step 3

Prove (m+1) is true.

We have to show that

$\:(m+1)(m+1+1)(m+1+5)$ is multiple of 3.

Then ,

$\sf\:(m+1)(m+2)(m+6)=(m^2+3m+2)(m+6)$

$\sf\:=m(m^2+3m+2)+6(m^2+3m+2)$

$\sf\:=m^3+3m^2+2m+6m^2+18m+12$

$\sf=m^3+9m^2+20m+12$

Put the value of Equation (1)

$\sf=3\lambda+3m^2+15m+12$

$\sf=3(\lambda+m^2+5m+4)$ , which is a multiple of 3

Therefore , p(m+1) is true.

Hence , by principal of mathematical induction , p(n) is true , where n is natural no.