Proof by Mathematical Induction
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:
- Proposition – What are you trying to prove?
- Base case – Is it true for the first case? This means is it true for the first possible value of .
- Assumption – We assume what we are trying to prove is true for a general number. such as , also known as the induction hypothesis.
- Induction – Show that if our assumption is true for the ( term, then it must also be true for the ( term.
- Conclusion – Formalise your proof.
There will be four types of mathematical induction that you will come across in FP1:
- Summation of series;
- Recurrence relations;
Example of a proof by summation of series edit
Example of a proof by divisibility edit
Notice our parameter, . This means that what we want to prove must be true for all values of which belong to the set (denoted by ) of positive integers ( ).
Assumption (Induction Hypothesis): Now we let where is a general positive integer, and we assume that .
Remember that .
Induction: Now we want to prove that the term is also divisible by 4
This is where our assumption comes in, if then 4 must also divide .
Now we've shown and thus . This implies that because you have successfully shown that 4 divides , where is a general, positive integer ( ) and also the consecutive term after the general term ( )
Example of a proof by recurrence relations edit
Example of a proof by matrices edit