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Find the sum

needs to be a whole number

sum=


or

take 1+2+...+n-1+n =

take half of n then add half

then take yor number after that multipy

(n/2+1/2)n

Proof:

if x = sum (1...n) then

x + (n + 1) = n * (n+1) / 2 + (n + 1)

x + (n + 1) = (n * (n + 1) + 2n +2) / 2

x + (n + 1) = ((n*n + n) + 2n + 2) / 2

x + (n + 1) = (n*n + 3 *n + 2) / 2

x + (n + 1) = ((n + 1) * (n + 2)) / 2


What if it doesn't start at 1?

Like 15+16+17+...

Minus the smaller consecutive numbers from the bigger ones.

Try from 21 to 80.

(80+1) * (80/2) - (20/2) * (20+1)

=(81 * 40) - (10 * 21)

=3240 - 210

=3030

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