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Pattern Recognition

In each of these puzzles, you'll try figure out what creative patterns we intended. However, if you're up for a challenge, try to then come up with a different, crazy way to explain each sequence! See more

Level 3

         

\[\large 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, \ldots \]

What is the 1000th term in the sequence above?

The above shows the first 4 pentagonal numbers, namely 1, 5, 12, 22.

How many points need to be added to the \(36^\text{th}\) pentagonal number to get the \(37^\text{th}\) pentagonal number?

The picture above illustrates which of the following:

Which is the missing square ? Assume that "blank" added to anything else results in a "blank".

\[\begin{array} & & & & 1& & \\ & & 2 & 3 & 4 & & \\ & 5 & 6 & 7 & 8 & 9 & \\ 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ &&& \vdots &&& \end{array}\]

Given that the pattern continues, find the second term in the \(13^\text{th}\) row.

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