n(n+1)(n+2) is a multiple of 6 .... Using mathematical induction prove this Share with your friends Share 6 Neha Sethi answered this Dear student Let P(n) be the statement "nn+1n+2 is a multiple of 6", i.eP(n):nn+1(n+2) is a multiple of 6STEP IWe have, P(1):11+11+2=6 is a multiple of 6.∴P(1) is true.STEP IILet P(m) be true. Then,m(m+1)(m+2) is a multiple of 6⇒m(m+1)(m+2)=6λ for some λ∈N. ....(1)Now, we shall show that P(m+1) is true. For this we have to show thatm+1m+2m+3 is a multiple of 6.We have,m+1m+2m+3=m+1m+2m+1+2=m+1m+2m+1+2m+1m+2=mm+2m+1+m+2m+1+2m+1m+2 =6λ+3m+1m+2Now, (m+1) and (m+2) are consecutive integers, so their product is even.Then, m+1m+2=2q evenm+1m+m+3=6λ+3(2q)=6(λ+q) which is a multiple of 6.⇒P(m+1):(m+1)(m+2)(m+3) is a multiple of 6⇒P(m+1) is true whenever P(m) is trueThus, P(1) is true and P(m+1) is true whenever P(m) is true.Hence by the principal of mathematical induction,P(n) is true for all n∈N. Regards 4 View Full Answer Jainamshah answered this First try by putting n=1 Then n=k And then put n=k+1 And solve it -25
