This question is for sequences, how would I go about solving this?

Question

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

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

Answer 1

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 $$a_n = \frac 1{2^n} = \left(\frac 12\right)^n.$$

Answer 2

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.