Can you explain mathematical induction?
Mathematical induction is a very important tool in maths. By this we can mathematically prove a statement. But there are certain limits in mathematical induction. This can only be used in positive integers. By proving the parts of statement one by one in a sequence we can prove the whole statement mathematically.
There are three important stages in mathematical induction.
- Prove the statement for n=1
- Assume at n=p (where p is a positive integer) the statement is true.
- Using this assumption prove that for n=p+1 the statement is true.
If p=2 then p+1 = 3 or if n=3 then p+1 =4
So like this we can produce the sequence of the statement is true.
So this sequence yields for all positive `n in Z` the statement is true.
Note that some special occasions are there that the starting point is not n=1. There may be some other point. But still the mathematical induction can be done for that starting value. Same thing can be found in n=p and n=p+1 terms. Some time when n=p the next point will be n=p+2 according to the requirements of the statement.
Show that f(n) is divisible by 5 when n is even.
So we can not take n=1 as starting value here because 1 is not even. So we have to prove the statement for n=2 first.
So if n=p where p is even for above example we cannot prove the result for n=p+1 because it is a odd number. So we have to go to n=p+2