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Double An Odd Sum

Problem

Consider the following results:

4 (1)
 = 1 + 3
4 (1 + 3)
 = 1 + 3 + 5 + 7
4 (1 + 3 + 5)
 = 1 + 3 + 5 + 7 + 9 + 11
4 (1 + 3 + 5 + 7)
 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15

If $S_n$ represents the sum of the first $n$ odd numbers, prove that $4S_n = S_{2n}$.

Problem ID: 283 (23 Jul 2006)     Difficulty: 2 Star

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