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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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