Induction

Mathematical Induction Example 5 --- Divisible by 3

Problem: For any natural number n , n³ + 2n is divisible by 3.

Proof:
Basis Step: If n = 0, then n³ + 2n = 0³ + 2*0 = 0. So it is divisible by 3.
Induction: Assume that for an arbitrary natural number n,
n³ + 2n is divisible by 3. -------- Induction Hypothesis
To prove this for n+1,   first try to express ( n + 1 )³ + 2( n + 1 )   in terms of n³ + 2n   and use the induction hypothesis.
( n + 1 )³ + 2( n + 1 ) = ( n³ + 3n² + 3n + 1 ) + ( 2n + 2 )
= ( n³ + 2n ) + ( 3n² + 3n + 3 )
= ( n³ + 2n ) + 3( n² + n + 1 )
which is divisible by 3, because ( n³ + 2n ) is divisible by 3 by the induction hypothesis.

End of Proof.