Applications based on the problem, "Sum of first n odd numbers"

Recall that, 

The sum of the first n odd numbers is n x n = n squared.

So, what if one wants to find the sum 11 + 13 + 15 + ... + 41 ? How do we do it?

Adding the odd numbers would be a tedious job. Not recommended.

Why not we try to use our earlier result(shown in orange)? In fact, we can use it very effectively.

We can write 11 + 13 + 15 + ... + 41 = (1 + 3 + 5 + ...+ 41) -  (1 + 3 + ... + 9 )

                                                                    =  21 x 21 - 5 x 5

                                                                    = 441 - 25 

                                                                    = 416

Notice that we need to determine how many odd numbers are there between 1 and 41 (both inclusive )and also between 1 and 9. That should not be difficult at all since we know that in the 1st 10 natural numbers, 5 are odd and the remaining 5 are even. Thus, in 40 natural numbers we have 4 x 5 = 20 odd numbers plus 1( to account for the number 41). Thus giving us a total count of 21. Using the same reasoning, we have 5 odd numbers between 1 and 9 (both inclusive).

Generate your own applications based on this formula and have fun.