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Find a general term (expressed as a function of the variable $n$) for the following sequence:

$$\{a_1, a_2, a_3, a_4,\dots\}=\left\{\frac{4}{8}, \frac{16}{64}, \frac{64}{512}, \frac{256}{4096},\;\ldots\right\}.$$

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  • $\begingroup$ If this account also belongs to you, you can merge the two accounts using the instructions here. (I am just guessing, based on the same username.) $\endgroup$ Feb 9, 2015 at 14:19

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You essentially have the sequence $$\{a_1, a_2, a_3, a_4,\ldots\} = \left\{\frac 12,\;\frac 14, \;\frac 18,\; \frac{1}{16},\; \ldots\right\}. $$ So the $n^{\large \text{th}}$ term in the sequence is given by $$a_n = \frac 1{2^n} = \left(\frac 12\right)^n.$$

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  • $\begingroup$ wow thanks can't believe I missed that $\endgroup$
    – Guest Anon
    Feb 8, 2015 at 16:36
  • $\begingroup$ You're welcome! $\endgroup$
    – amWhy
    Feb 8, 2015 at 16:40
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Sometimes it's helpful to treat the numerator and the denominator as separate sequences.$$\{4,16,64...\}\implies a_n=4^n \\\{8,64,512..\}\implies b_n=8^n$$ So the nth term of the original sequence is $c_n=\frac{a_n}{b_n}=\frac{4^n}{8^n}=\left(\frac{1}{2}\right)^n$

I thought this might be useful to add for situations that are difficult to spot.

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