1000?

Algebra Level 1

Calculate: 1+2+3+4+.......1000.


The answer is 500500.

Fred Fan by fred fan , 21, Canada

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1 solution

Caleb Townsend
Apr 9, 2015

The sum of the first n n positive integers is T n = n ( n + 1 ) 2 T_n = \frac{n(n+1)}{2} Substitute n = 1000 n=1000 and T n = 500 × 1001 = 500500. T_n = 500\times 1001 = 500500.
The formula can be proven by induction quite easily, or with A.P. formula. There are also formulae for the sum of squares, cubes, fourth powers, and so forth, but they grow in complexity.

Alternatively, you could use a method Gauss is said to have used as a small child. 1 + 2 + 3 + . . . + 999 + 1000 = ( 1 + 1000 ) + ( 2 + 999 ) + . . . + ( 500 + 501 ) = 500 × 1001 1 + 2 + 3 + ... + 999 + 1000 \\= (1 + 1000) + (2 + 999) + ... + (500 + 501) \\ = 500\times 1001

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Do you know this pattern?

The sum of integers from 1 to 10 is 55

The sum of integers from 1 to 100 is 5050

The sum of integers from 1 to 1000 is 500500

The sum of integers from 1 to 10000 is 50005000

...

Thus the sum of integers from 1 to 10^n is ever (10^n)/2 wrote 2 times together!

Victor Paes Plinio - 6 years, 1 month ago

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