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 \(=\) \(1 + 3 + 5 + 7 + 9 = 25 = 5^2\)
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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 \(n^2\).
\(n^{th}\) odd number:
The sequence of odd numbers is: \(1, 3, 5, 7, 9, 11, ...\)
For \(n =1\), \(2(1) -1 =1\)
For \(n =2\), \(2(2) -1 =3\)
For \(n =3\), \(2(3) -1 =5\)
For \(n =4\), \(2(4) -1 =7\)
For \(n =5\), \(2(5) -1 =9\)
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Thus, \(n^{th}\) odd number is \(2n - 1\).