Question

Use mathematical induction to prove the statement is true for all positive integers n.

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

Answer 1

Did you reupload the question? I was posting the answer to the earlier post but it suddenly said that the question is closed so I just stopped midway

Answer 2

My computer shut off and when I went to go back to it I couldn't find it, I'm sorry.

Answer 3

ok no big deal. so I'm not really used to this 'mathematical induction' but I proved it by dividing the "n" into 3 possibilities of 3p/3p+1/3p+2

Answer 4

we can agree that ALL integers can be described by it being one of the three (p also being an integer)

Answer 5

so back to your original formula; n^3 + 2n = n(n^2 + 2)

Answer 6

if n = 3p: 3p(9p^2 + 2); this is divisible by 3 because of the 3p

Answer 7

I'm not to good at this math induction either

Answer 8

if n = 3p+1: (3p + 1)(9p^2 + 6p + 1 + 2) which is also (3p+1)(9p^2+6p+3). the second parenthesis, we can take a 3 out which gives us 3(3p+1)(3p^2+2p+1), which is also divisible by 3

Answer 9

lastly if n = 3p+2: (3p+2)(9p^2 + 12p + 4 + 2) which is (3p+2)(9p^2+12p+6); the second parenthesis, we can take a 3 out again which gives us 3(3p+2)(3p^2+4p+2) which is divisible by 3

Answer 10

thus we counted all the possibilities of n.

Answer 11

Thank you for helping me.