Question

for every positive integer is N Cube prove that n cube-1 is divisible by 6​

Answer 1

Step-by-step explanation:

Answer. By the mathematical induction we have to prove this. ⇒ Given that S(n) = n3-n divisible by 6. Let n =1 then we get '0' which is divisible by 6. ∴ S(1) is true. Let us assume that n = k S(k) = k3- k which is divisible by 6. ∴ S(k) is true. ∴ (k3-k) / 6 = m ( integer ) (k3-k) = 6m k3= 6m +k --------→(1) now we have to prove that n = k+1 ⇒ (k+1)3 - (k+1) ⇒ (k3+3k2+3k+1) - (k+1) subsitute equation (1) in above equation then ⇒ 6m +k+3k2+2k ⇒ 6m +3k2+k ⇒ 6m +3k(k+1) ( ∴k(k+1) = 2p is an even number p is natural number) ⇒ 6m +3x2p ⇒ 6(m +p) ∴which is divisible by 6 s(k+1) is true. By the mathematical induction it is true for n∈N.