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Mental math- The sum of the first $50$ positive odd integers is $50^2$. Find the sum of the first $50$ positive even integers.

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Hint: each of these even integers is one bigger than one of the odd integers. –  user108903 Dec 16 '12 at 20:18
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There are $50$ kids lined up, you give them $1$ candy, $3$, $5$, and so on. Total candies $50^2$. Now you give each kid one more. –  André Nicolas Dec 16 '12 at 20:21
    
homework should not be used as a standalone tag; see tag-wiki and meta. –  Martin Sleziak Dec 16 '12 at 21:00
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1 Answer

You are given that $$1 + 3 + 5 + \cdots + 99 = 50^2$$ You need to compute $$2 + 4 + 6 + \cdots + 100$$ Note that \begin{align} 2 + 4 + 6 + \cdots + 100 & = (1+1) + (3+1) + (5+1) + \cdots + (99+1)\\ & = (1 + 3 + 5 + \cdots + 99) + (\underbrace{1 + 1 + 1 + \cdots + 1}_{50 \text{ times}})\\ & = 50^2 + 50 \end{align}

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