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hannahr

Guest

another induction

How would one show this by induction?

   is an integer given a and b are natural numbers

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: another induction

Hi hannahr,

Do you know 'Pascals Triangle'.  This can be used to generate terms in binomial expansions

eg.   

The expression

is the formula for the coefficients in this expansion.

So you can use the Pascal rule to do the induction step.

Consider the expression

This is the general term for the next row down in the triangle.

split off some bits from the factorials

separate into two fractions

cancel

multiply out

This shows that a term in the triangle is made by adding the two terms in the row above.

If they are each integers, then their sum will be too.

So if any row of the triangle consists of all integers, then the row below will be too.

Finally you need an initial step.

Show that

is true when a = 1 and b = 1 and you have an inductive proof.

Hope that helps.  Post again if you want more on this.

Bob

Last edited by Bob (2010-08-23 23:59:53)

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