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Sequences and Series

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

Problem Solving

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

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

If \(\{a_n\}\) is a geometric progression with \(a_{3}=\frac{1}{4}\) and \(a_{7}=\frac{1}{64},\) what is the smallest integer \(n\) such that \(a_n <\frac{1}{1000}?\)

Sequence \(\{a_n\}\) satisfies \[a_1=3, a_{n+1}=\frac{n}{n+1} a_n,\] where \(n\) is a positive integer. What is the value of \(a_{133}\)?

Sequence \(\{a_n\}\) satisfies \[a_2=2a_1, a_{n+2}-4a_{n+1}+3a_n=0,\] where \(n\) is a positive integer. If \(a_5=287\), what is the value of \(a_4\)?

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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