Basic Mathematics

Pattern Recognition

Pattern Recognition: Level 3 Challenges

         

1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,\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 36th36^\text{th} pentagonal number to get the 37th37^\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".

12345678910111213141516\begin{array}{c}& & & & 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 13th13^\text{th} row.

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