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.

Preface

Dear Blog Reader,

I thought of designing this blog with the intention of facilitating better understanding of Mathematics and as a result enjoying the very process of learning. This blog will not be restricted to any standard or level as such but whatever the level will be mentioned except for General Mathematics. This blog can be happily used by students, parents, teachers and all mathematics enthusiasts. Feel free to post your comments and any discrepancy you may find therein. It may take some time to settle to an accepted design which will be modified over the course of time as more and more posts appear.

Julius Fernandes

Let me start with a problem:

What is the sum of the first 10 odd numbers?

Solution: In order to find the sum of the first 10 odd numbers, we need to list those odd numbers. An odd number is a number which is not divisible by 2. In other words, it is not even. 

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, ...

Since we need only the first 10 of them, we stop at number 19.

Hence, 1 + 3 +5 + 7 + 9 + 11+ 13 + 15 + 17 + 19 = 100.

So This is how a traditional thought process would function. We can also choose to see it in a different way. Suppose, we add the numbers in the following manner,

1, 

1+3,

1+3+5,

1+3+5+7,

1+3+5+7+9,

1+3+5+7+9+11,

1+3+5+7+9+11+13,

1+3+5+7+9+11+13+15,

1+3+5+7+9+11+13+15+17,

1+3+5+7+9+11+13+15+17+19,

We now add each of these groups of numbers.

1=1, 

1+3=4, 

1+3+5=9, 

1+3+5+7=16, 

1+3+5+7+9=25, 

1+3+5+7+9+11=36, 

1+3+5+7+9+11+13=49, 

1+3+5+7+9+11+13+15=64,

1+3+5+7+9+11+13+15+17=81,

1+3+5+7+9+11+13+15+17+19=100,

Now observe each of the sum of these carefully and check whether there evolves a pattern that we can possibly use.

1=1=1x1, 

1+3=4=2x2, 

1+3+5=9=3x3, 

1+3+5+7=16=4x4, 

1+3+5+7+9=25=5x5, 

1+3+5+7+9+11=36=6x6, 

1+3+5+7+9+11+13=49=7x7 

1+3+5+7+9+11+13+15=64=8x8,

1+3+5+7+9+11+13+15+17=81=9x9,

1+3+5+7+9+11+13+15+17+19=100=10x10.

Thus, we see that when we add the first 10 odd numbers, it is 100 which is the square of the number of odd numbers taken.

If we were to take a finite number of odd numbers in order then the sum(addition) of all of them is the square of number of odd numbers taken.

i.e. 1+3+5+7+...+(2n-1)= n x n or n-squared. [n is a natural number].

So, if one is curious to know the sum of the first 15 odd numbers it will be 15x15=225 and so on. Try it.