SOLUTION: 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.

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Question 1135570: 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 by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
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Let’s prove it using principle of mathematical induction (PMI).


    P(n)=n^3+2n.


For n=1,

    P(1)=1+2=3 which is divisible by 3.

so the base of induction is established.


Now for n=k, assume that

    P(k)=k^3+2k

is divisible by 3.


Then for n=k+1,

    P(k+1)=(k+1)^3+2(k+1) = k^3+2k+3k^2+3k+3=P(k)+3(k^2+k+1)

Since we assumed P(k) to be divisible by 3, therefore P(k+1) is also divisible by 3.


Hence by PMI,  n^3+2n  is divisible by  3  for any integer positive n.

The proof is completed.