# Adding Consecutive Numbers

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The count is easy and try but there is a faster shortcut.

You can count 1+2+3+4+5+6+... tiring right.

There a shortcut for it is to take as 1+2+3+4+...+n

First take n/2

next take n+1

then mutiply

(n/2)(n+1) is 1+2+3+...+n

n is an algebrabic unit.

try 10

10/2=5

10+1=11

5*11=55

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