Playing With Numbers: Triangles and Squares

You can play with numbers; which will be a surprise to some and extremely obvious to others.  I'm writing for those who will be surprised.  Consider the picture of dots here:
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We've got a triangle, a small one.  It has 3 dots.  Now put another row of dots, keeping it a triangle:
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There are 3 dots in the first triangle, 6 in the second.  Next triangle will have 10 (as we add in a row of 4). 

For gaming: What is the 20th triangle number?  Is there a way you can look at a number and tell whether it is triangular?

Or you can play with squares:
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So the first three square numbers are 1, 4, 9.  Next, the 4th square number, will be 16.  These are actually simpler to game than the triangular numbers.  What's the 20th square number?

And of course we can make more interesting figures, like hexagons:
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So the first hexagonal number is 7.  What's the second?  Can you predict the 3rd, the 20th?

On the one hand, we're just playing some games here.  On the other, there are also serious mathematical papers on hexagonal numbers, and triangular, octagonal, and so forth.