SOLUTION: What is the rule for the sum of the first 'n' positive integers. I.e. 1+2+3+4+5... Could you please give the formula and explain the solution?

Question 853347: What is the rule for the sum of the first 'n' positive integers.
I.e. 1+2+3+4+5...
Could you please give the formula and explain the solution?
Found 2 solutions by richard1234, rothauserc:
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
1 + 2 + 3 + ... + n = n(n+1)/2

Can be proven by induction, or by noting that

S = 1 + 2 + ... + n
S = n + (n-1) + .. + 1
------------------
2S = (n+1) + (n+1) + ... + (n+1) = n(n+1) --> S = n(n+1)/2
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
the sum of the first n positive integers is given by the formula
n(n+1) / 2
note that if n = 1, then
1(1+1) / 2 = 2/2 = 1
now assume that the formula
n(n+1) / 2 is true for n > 1
we need to show that
sum of i where i =1, n+1 is (n+1)(n+2) / 2
pull n+1 out of the sum and we get
sum of i where i =1, n+1 is n+1 + sum of i where i =1,n is (n+1)+ n(n+1)/2
factor out (n+1) and simplify
(n+1) + n(n+1) / 2 = (n+1)(1+ n/2) = (n+1)(2+n)/2 = (n+1)(n+2) / 2
which proves the result

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