Mathematical approach to determine Sum of natural number and sum of odd natural number

sum of natural number
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s  = 1 + 2     + 3     + 4     + . . . . . . + n
s  = n + (n-1) + (n-2) + (n-3) + (n-4) + ..
2s = (n+1) + (n+1) + (n+1) + (n+1) + (n+1) . . . . 
s  = ((n+1) + (n+1) + (n+1) + (n+1) + (n+1) . . . . ) / 2
s = (n(n+1)) / 2

sum of odd natural number
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s = 1      + 3      + 5        + 7        +..... + (2n-1)
s = (2n-1) + (2n-3) + (2n - 5) + (2n - 7) + ....
2s = 2n + 2n + 2n + ......
2s = n * 2n 
s = n * 2n / 2
s = n * n
s = n**2

Sum of natural number in PHP

Sum of odd natural number in PHP