In simple terms Proof of Mathematical Induction is a process typically which is used for proving of the Math statement which is true for all positive integers. Perhaps we must not get confused with the proof by mathematical induction with the non-rigorous reasoning of mathematics, whereas it is a rigorous deductive reasoning of mathematics. We can understand this by taking an example;
Sum of Odd NumbersBack to Top
1: First property of odd numbers: When we add two odd numbers then sum is always equals to an even number. For example:
1+ 3 = 4,
4 is an even number.
When we add even number of odd numbers we get resulting sum as an even number. For example:
3 + 5+ 7+ 9 = 24
2: When we add odd number of odd numbers then we get resulting sum as an odd number. For example:
1 + 3 + 5 = 9
9 is also an odd number.
3: There is a special property of sum of odd numbers that when we add first 'n' odd numbers then result is always a perfect Square number. For example:
1 + 3 = 4,
1 + 3 + 5 = 9,
4 and 9 both are perfect square numbers.
4: If we add first 'n' odd numbers which has the difference of 2 then we can calculate the sum of odd numbers by formula n2.
Sum of 'n' odd numbers = n 2.
If difference d = 2.
5: We can represent the sum of 'n' odd numbers by equation:
1 + 3 + 5 + 7 + . . . . . . + 2n – 1 = n 2.
This equation is a geometric interpretation of sum of odd numbers.
Counting RegionsBack to Top
Let us derive a formula to know maximum number of regions that are possible to be made by 'N' straight lines residing in a plane. Suppose we have 'N' lines in a plane with no line parallel to each other and so there are possibilities of their intersections. The regions that are formed by the Intersecting Lines can be evaluated as:
Based on our counting we find these number of regions being formed by each line as shown in the above figure:
Line 1: 2 regions
Line 2: 4 regions
Line 3: 7 regions
Line 4: 11 regions
Arranging these region counts in a particular order we see that they follow a particular fashion. Our sequence is 2, 4, 7, 11. By this we can easily derive the general formula for counting the regions for any number of lines we consider. The formula is:
Σ (N + 1) = (N (N + 1) / 2) + 1,
We have many applications of this concept in mathematics especially in geometrical problems. Let us see an example of it.
Example: In the following figure find out the total number of squares:
Solution: In figure we can see that there are 4 sides possible. Calculating number of squares for each side we get:
Number of squares that can be seen of side 1 is equals to 4 * 6 = 24. Likewise we have 15 squares of side 2, 8 of side 3 and 3 of side 4. Summing up all of them we get 50 squares in total in the given figure.