New way to find the Sum of natural numbers
Posted on: April 26, 2025
For any positive integers a, b, c, d, e:
S(a + b + c + d + e ... ) = {Sum of the all combination product of the given variables} + S(a) + S(b) + S(c) + S(d) + S(e) + ..
For 3 variable equation:
S(x + y + z) = xy + yz + zx + S(x) + S(y) + S(z)
where S(n):
\[ S(n) = \frac{n(n+1)}{2} \]
Example:
Let x = 5 , y = 4 , and z = 8 :
-
\[ (5 \times 4) + (5 \times 8) + (4 \times 8) = 20 + 40 + 32 = 92 \]
-
S(5) = 15, S(4) = 10, S(8) = 36
-
15 + 10 + 36 = 61
- Total:
92 + 61 = 153
- Also:
\[ S(5+4+8) = S(17) = \frac{17\times18}{2} = 153 \]
We can use any number of variables in any order.