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What's the sum of the first 100 positive integers? How about the first 1000?

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Suppose \(\{a_n\}\) is a sequence defined by \[a_1 = 1, a_2 = 1,\] and \[a_n = a_{n-1} + a_{n -2}\] for each natural number \(n > 2.\)

What is \(a_6\)?

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A *geometric progression* is a sequence in which \(a_n = r \cdot a_{n-1}\) for each natural number \(n > 1\), where \(r\) is a real number called the *common ratio*.

If \(a_n\) is a geometric progression with \(a_1 = 5\) and \(a_6 = 160\), what is \(a_3\)?

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