# Recognizing the sequence 1/16, 1/8, 3/16, 1/4, 5/16, …

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

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OMG, of course. Thank you for the heads up. –  user1495024 Aug 2 at 1:22
Am I the only one who thought $\frac{1}{2}$ before reading the possible options? :) –  Thomas Aug 2 at 4:15
No, the next two numbers should definitely be 1/2 and 7/16 :-) –  CompuChip Aug 2 at 8:23
@Thomas I did consider it, but then thought it wasn't just right. First of all, I'd be generalizing from just two data points. Secondly I'd be ignoring the other three data points, which supposedly were there for a reason. –  kasperd Aug 2 at 9:37

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

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

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

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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 $\frac{5}{16}+\frac{1}{16} =\frac{6}{16} =\frac{3}{8}$.

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