I can't understand the logic here. Please help.

https://brilliant.org/practice/divisibility-visual/?p=9

The problem is from the link above. Since I don't understand almost every possible sentences from that page, I'm very troubled for what I should even ask.

Firstly, how is it possible for the triangular number with 3k+1 as the last number to be followed by the one with 3k+3 as its last number? As you can see on the example above, 3k = 6, then 3k+1 = 7, which makes sense since the next triangular number for 21 is 21+7, 28. But how could 30, 21+9, where 9 come from 3k+3, be the second next triangular number? Is it perhaps that 3k in 3k+1 is a different number with 3k in 3k+3?

Secondly, how can they be sure they can show the pattern continues just by showing 3 cases?

And, in the last pictures, how does it make sense to say the first two figures are triangular numbers? They just aren't, judging from the look of them? It's not stair-like figure at all.

#Algebra

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
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`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$