SOLUTION: ) Let the sequence (an) be defined by an+1 = an + 3n for n ≥ 1 and a1 = 6. Prove by induction that an > 5.

This problem certainly does not "need" induction to prove b/c the terms are monotonically increasing and they start with  so all  ... but here is the inductive proof:



Base case:

  6 > 5, so the base case is true



Hypothesis:

 for n=k



we also can write

  (*)



Step case:

Let n=k+1.  We must show  for n=k+1 (then it holds for all k>=1)





Substitute by using (*):





Now,   > 5 by the hypothesis  (and  3(k+1) > 0  since k>0) 

so the RHS is greater than 5 and  for n=k+1.



Therefore  for  ■





In my opinion, much better introductory problems exist for teaching proof by induction.