Formula to find sum of the first n Natural numbers
Hello, Everyone. In this, we are going to see a proof for the formula to find sum of the first n Natural numbers. Let S = 1 + 2 + 3 + 4 + ......... + (n-1) + n , where n is any natural number. --------------- (1) The r th term from the starting of the series, in the above series is r. ------- (2) From the property of addition of numbers, we can write S as shown below. S = n + (n-1) + (n-2) + ......... + 2 + 1 ---------- (3) The r th term from the starting of the series, in the above series is n-(r-1) = n - r + 1 -------- (4) Let us add (1) and (3) such that the corresponding terms are added as shown in the below figure. From (2) and (4), The r th term from the starting of the series, in the resulting series obtained from addition of (1) and (3) is equal to r + (n - r + 1) = n + 1. From this, it is obvious that the r th term of the resulting series is independent of r. Therefore, as n is a constant, every term in the resulting series is equal to n + 1. Therefore, by add...
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