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Resources » Articles/Knowledge Sharing » Education »
Mathematical Induction
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Hello Friend,
In our daily life, we use various kind of reasoning depending on the situation we are faced with. For instance, if you are told that your friend has just a child, you would know that it is either a girl or a boy. In this case you would be applying general principles to a particular case. This form of reasoning is an example of deductive logic.
Now let us consider another situation. When you look around you are find students who study regularly do well in exams, you may formulate the general rule ( rightly or wrongly ) that " any one who studies regularly will do well in examinations." In this case you would be formulating a general principle or rule based on several particular instances. Such reasoning is inductive, a process of reasoning by which general rules are discovered by the observation and consideration of several individual cases. Such reasoning is used in all the sciences, as well as Mathematics.
Mathematical induction is a more precise form of this process. This precision is required because a statement is accepted to be true mathematically only if it can be shown to be true for each and every case that it refers to.
Mathematical Induction : -
Let P(n) be a statement involving a natural number n. If
- It is true for n=1, i.e. P(1) is true ; and
- Assuming k>=1 and P(k) to be true it can be proved that P(k+1) is true ;
then P(n) must be true for every natural number n.
The condition above does not say that P(k) is true. It says that whenever P(k) is true, then P(k+1) is true.
For better understanding you can consider the following example.
Example : -
Prove that 1 + 2 + 3 + 4 +....................+n = (n/2)(n+1)
Solution : -
- for n=1 ;
1=(1/2)(1+1) ;
1=1 ;
- Let P(k) is true thus,
1 + 2 + 3 + 4 +....................+ k = (k/2)(k+1);
Now for P(k+1) ;
1+...........+k+(k+1)=[(k+1)/2](k+2) ;
by simplifying we get
L.H.S=R.H.S
This shows that the above said equation is valid for each integer.
- Article by,
Sunil Saharan
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