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# Sequences and Series

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

\[ \frac11, \frac21, \frac12, \frac31, \frac22, \frac13, \frac41, \frac32, \frac23, \frac14, \cdots \]

If the \(n^\text{th}\) term of the above sequence is \( \frac{3}{15} \), what is \(n\)?

A sequence \( \{a_n\} \) with \(a_n>0\) for all \(n(\ge 1)\) satisfies

\[ \frac{a_{n+1} - a_n}{a_{n+1} + a_n} = \frac12.\]

If \(a_1 = 2\), what is \(a_5\)?

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