Mathematical Induction

Consider the inequality :

Can you use Mathematical Induction to prove this?

Please make a try before you carry on.

Got stuck!

Most people have a habit of taking up a question and carrying out their work at once. They may try different methods to tackle the question.

But they are afraid to make changes or adjustments to the question itself.

Now let us change the question to a new one, as shown below.

P(1) : L.H.S.

R.H.S.

\ L.H.S.£ R.H.S.

\ P(1) is true.

Assume P(k) is true for some positive integer k,

i.e. (*)

For P(k+1),

, by (*)

, since k < k+1

\ P(k+1) is also true.

By the Principle of Mathematical Induction, P(n) is true for all positive integers, n.

Back to our original question

Note that it is not difficult to prove:

Now,

And the proof completes!

Finale : How can we figure out the need to change the question to:

Find out yourselves.

Lesson : Question the question! - K.C.Yue