What is the sum of the first thousand positive odd numbers?

What is the sum of the first thousand positive odd numbers?

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Solution

The first odd number in the list is one. The n^th odd number is given by $2n-1$, so the thousandth is 1999. We can use the sum of an arithimetic sequence, $$S_n=\frac{n}{2}[a_1+a_n]$$ to determine the sum. So this sum is $$S_{\mathrm{1000}}=\frac{1000}{2}[1+\mathrm{1999}]$$ $$=500\times 2000$$ $$=\mathrm{1,000,000}$$

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