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4. Interesting patterns in square numbers

Summing up odd natural numbers:

Odd numbers are \(1\), \(3\), \(5\), \(7\), \(9\), \(11\), \(13\), \(15\), ...

First odd number \(=\) \(1 = 1^2\)

Sum of first two odd numbers \(=\) \(1 + 3 = 4 = 2^2\)

Sum of first three odd numbers \(=\) \(1 + 3 + 5 = 9 = 3^2\)

Sum of first four odd numbers \(=\) \(1 + 3 + 5 + 7 = 16 = 4^2\)

Sum of first five odd numbers \(=\) \(1 + 3 + 5 + 7 + 9 = 25 = 5^2\)

….

Sum of first \(n\) odd numbers \(=\) \(1 + 3 + 5 + 7 + 9 + 11 + … = n^2\)

The sum of the first \(n\) consecutive odd natural numbers is \(n^2\).

Summing up two consecutive natural numbers results in an odd square:

Let us take any odd natural number, say \(9\).

The square of \(9\) is \(81\).

To find the sum of two consecutive natural numbers equal to the odd square, we can use the below formula.

a^{2} = \frac{a^{2} - 1}{2} + \frac{a^{2} + 1}{2}

Now, let's take \(a = 9\).

Substitute the value of \(a\) in the above formula.

9^{2} = \frac{9^{2} - 1}{2} + \frac{9^{2} + 1}{2}

81 = \frac{80}{2} + \frac{82}{2}

\(81 = 40 + 41\)

The numbers \(40\) and \(41\) are consecutive.

Therefore, \(81\) can be written as the sum of \(40\) and \(41\).

Important!

The square of any odd number can be written as the sum of two consecutive natural numbers.

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