How Odd Numbers Combine to Form Square Numbers?

Numbers were first introduced for counting. When mathematics began its development it became more than counting. Many number theories have been formulated and even some people spend their whole life researching on numbers and their properties. Either it may be the simple natural numbers that our grandparents use for counting goats and chicken or it may be the irrational and imaginary numbers consisting i, e, π that physicists, mathematicians or scientists use, all they carry mathematical beauty.

Mathematics is an entire world for them who feel it. Numbers are like we human. In the way we human are classified as male and female (pardon third gender) on the basis of sex, natural numbers are classified as odd and even numbers on the basis of divisibility by 2. If we go microscopic into numbers we can get into sea of classification. Here I will show one property of set of first consecutive odd numbers and explain its proof. It may look simple but believe me it is beautiful one.

Property:

If we sum the first certain consecutive odd numbers, we get the perfect square number.

Illustration:

Let us take set of first n consecutive odd numbers as:

A= {1, 3, 5, 7 … n}

I will form its sequence as:

S = 1 + 3 + 5 + 7 + 9 + ….n

Now our task is to take any n and find the sum(S).

N = 1 è  S = 1 = 1²

N = 2 è  S = 1 + 3 = 4 = 2²

N = 3 è  S = 1 + 3 + 5 = 9 = 3²

N = 4 è  S = 1 + 3 + 5 + 7 = 16 = 4²

And so on.

Here I obtained sum as a perfect square number. Now let’s see how it works:

I form a series S_n as sum of first n perfect square numbers.

So, S_n = 1² + 2² + 3² + 4² + …………. + n²

      S_n =         1² + 2² + 3² + 4² + …..... + (n-1)² + n²

Shifting the R.H.S. one term right and subtracting,

0=1 + 2² - 1² + 3² – 2² + 4² - 3² + … + n² – (n-1)² – n²

or,  n² =1 + (2-1)(2+1) + (3-2)(3+2) + (4-3)(4+3) + … + (n-n+1)(n+n-1)

or,  n² = 1 + 1.3 + 1.5 + 1.7 + … + 1.(2n-1)

So, n² = 1 + 3 + 5 + 7 + … (2n-1)

Hence, proved.

We can also write as,

1 + 3 + 5 + 7 + 9 + … + n = [(n+1)/2] ²   …………… (1)

If you became lucky and someone asked you to find sum of odd consecutive numbers from 1 to 99 or 49 or any number ending with 9 use the identity (1) and you could do it in no time and answer. CHEERS!

Its geometry goes as:

J     J     J     J     J                                                             J

J     J     J     J     J                                                         J J J

J     J     J     J     J             è                                   J J J J J

J     J     J     J     J                                                J J J J J J J

J     J     J     J     J                                             J J J J J J J J J

Now, it’s your work to relate it with area of triangle and parallelogram.

I do hope you enjoyed it.

Thank You.