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Sequences and Series

Sequences and Series - Problem Solving


11,21,12,31,22,13,41,32,23,14, \frac11, \frac21, \frac12, \frac31, \frac22, \frac13, \frac41, \frac32, \frac23, \frac14, \cdots

If the nthn^\text{th} term of the above sequence is 315 \frac{3}{15} , what is nn?

If {an}\{a_n\} is a geometric progression where all terms are positive and a3=14a_{3}=\frac{1}{4} and a7=164,a_{7}=\frac{1}{64}, what is the smallest integer nn such that an<11000?a_n <\frac{1}{1000}?

Sequence {an}\{a_n\} satisfies a1=3,an+1=nn+1an,a_1=3, a_{n+1}=\frac{n}{n+1} a_n, where nn is a positive integer. What is the value of a133a_{133}?

Sequence {an}\{a_n\} satisfies a2=2a1,an+24an+1+3an=0,a_2=2a_1, a_{n+2}-4a_{n+1}+3a_n=0, where nn is a positive integer. If a5=287a_5=287, what is the value of a4a_4?

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

an+1anan+1+an=12. \frac{a_{n+1} - a_n}{a_{n+1} + a_n} = \frac12.

If a1=2a_1 = 2, what is a5a_5?


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