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If you add all numbers from $1$ to $400$ how much will you get? $$1+2+3.....398+399+400=?$$

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    $\begingroup$ arithmetic sequence? $\endgroup$
    – Arnaldo
    Dec 15, 2016 at 18:54

3 Answers 3

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The best way to see why the arithmetic sequence formula (mentioned by others) works is by observing $$ 2( 1 + 2 + ... + 400 ) = (1 + 400) + (2 + 399) + (3 + 398) + ... + (400 + 1) = 401 + 401 + ... + 401 = 400(400 + 1).$$ Then divide both sides by $2$ to obtain. $$ 1 + 2 + ... + 400 = \frac{400(400 + 1)}{2}.$$

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Simply $$1+2+\dots+400=\frac{400(400+1)}{2}=80200$$

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Simple arithmetic progression: $$\frac{1+400}{2}\cdot400 \quad =\quad 80200$$

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