Question

The integer n3 + 2n is divisible by 3 for every positive integer n

prove it by math induction

is it my proof right ?

Answer 1

P(n) = n^3 + 2n is divisible by 3 for every positive integer n.
Let's show that P(n) holds for n = 1
P(1) = 1^3 + 2(1) = 1 + 2 = 3 which is divisible by 3.

Now assuming, that p(k) is true, let's show that p(k + 1) is also true
p(k + 1) = (k + 1)^3 + 2(k + 1) = k^3 + 3k^2 + 3k + 1 + 2k + 2 = k^3 + 3k^2 + 5k + 3 = k^3 + 2k + 3k^2 + 3k + 3 = k^3 + 2k + 3(k^2 + k + 1), since P(k) is true => k^3 + 2k is true and 3(k^2 + k + 1) is divisible by 3 for all values of k wich shows that P(k + 1)is also true.
Therefore, P(n) is true for all positive integer value of n.