Question

sow that n3+2n is divisible by 3 for all n>1 by induction

Answer 1

n^3 + 2n you mean?

Answer 2

Take n =2
then n^3+2n = 12 which is divisible by 3

Answer 3

Now assume n^3 + 2n is divisible by 3 for some value of n
and try to prove that (n+1)^3 +2(n+1) is also divisible by n if the above is true

Answer 4

(n+1)^3 + 2(n+1) = n^3 +1+ 3n^2 + 3n + 2n +2
= (n^3 +2n) +3(n^2 +n+1)

Answer 5

Now we assumed n^3 +2n is already divisible by 3
and the rest of the expression has 3 as its factor so it is obviously divisible by 3

Answer 6

The whole expression is divisible by 3. We already proved earlier that the above statement is true for n=2 and here we have proved that if the statement is true for n=2 it will be true for n=3,4,5 and so on also