Recognizing the sequence 1/16, 1/8, 3/16, 1/4, 5/16, ...

Question

What is the missing number? $$\frac{1}{16}, \frac{1}{8}, \frac{3}{16}, \frac{1}{4}, \frac{5}{16}, \ \ \ [?]$$

$$A. \frac{5}{4}\quad B. \frac{3}{4}\quad C. \frac{5}{8}\quad D. \frac{3}{8}$$

Spoiler: Answer is $D$, but I don't know why.

Thanks

Answer 1

$$\frac{1}{16}, \frac{1}{8}=\frac{2}{16}, \frac{3}{16}, \frac{1}{4}=\frac{4}{16}, \frac{5}{16}$$

So the $i$th term is of the form $$\frac{i}{16}$$ Therefore, the next term is $$\frac{6}{16}=\frac{3}{8}$$

Answer 2

Another sequence-recognizing technique is to look at the difference between consecutive terms.

In this case, $\frac{1}{8}-\frac{1}{16} = \frac{1}{16} $, $\frac{3}{16}-\frac{1}{8} = \frac{1}{16} $, $\frac{1}{4}-\frac{3}{16} = \frac{1}{16} $, and $\frac{5}{16}-\frac{1}{4} = \frac{1}{16} $.

Since the difference between consecutive terms is $\frac{1}{16}$, the next term should be $\frac{5}{16}+\frac{1}{16} =\frac{6}{16} =\frac{3}{8} $.

Answer 3

$$\frac{1}{16}, \frac{1}{8}, \frac{3}{16}, \frac{1}{4}, \frac{5}{16}$$ The above is the same as $\displaystyle\frac1{16},\frac2{16},\frac3{16},\frac4{16},\frac5{16}$.