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Proof by mathematical induction[1]Edit

Mathematical induction is the process of verifying or proving a mathematical statement is true for all values of   within given parameters. For example:

 

We are asked to prove that   is divisible by 4. We can test if it's true by giving   values.

     
     
     
     
     
     

So, the first 5 values of n are divisible by 4, but what about all cases? That's where mathematical induction comes in.

Mathematical induction is a rigorous process, as such all proofs must have the same general format:

  1. Proposition – What are you trying to prove?
  2. basis case – Is it true for the first case? This means is it true for the first possible value of  .
  3. Assumption – We assume what we are trying to prove is true for a general number. such as  
  4. Induction – Show that if our assumption is true for the (  term, then it must be true for the term after (  term.
  5. Conclusion – Formalise your proof.

There will be four types of mathematical induction you will come across in FP1:

  1. Summing series
  2. Divisibility
  3. Recurrence relations
  4. Matrices

Example of a proof by summation of seriesEdit

Example of a proof by divisibilityEdit

Proposition:  

Note our parameter,   This means it wants us to prove that it's true for all values of   which belong to the set ( ) of positive integers ( )

Basis case:

 

Assumption: Now we let   where   is a general positive integer and we assume that  

Remember  

Induction: Now we want to prove that the   term is also divisible by 4

Hence  

This is where our assumption comes in, if  then 4 must also divide  

So:  

 

Now we've shown   and thus   it implies   because you have successfully shown that 4 divides  , where   is a general, positive integer ( ) and also the consecutive term after the general term ( )

Conclusion:

 

 

ReferencesEdit