SOLUTION: I have a problem with a sequence of dots. When n=1, there's one dot. When n=2, there's a shape that looks like a 2x2 rectangle made of 4 dots, and a triangle with 1 dot on top form

Question 1096056: I have a problem with a sequence of dots. When n=1, there's one dot. When n=2, there's a shape that looks like a 2x2 rectangle made of 4 dots, and a triangle with 1 dot on top forming a "house looking shape" with 5 dots. When n=3, it's a 3x3 rectangle with a triangle with 3 dots. I think it might be easier to upload an image: https://imgur.com/a/g9UOx
Thank you!
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

You didn't say what you are supposed to do with this pattern.... I would guess you are looking for a formula for the number of dots in the n-th figure. Let's call the formula for the number of dots in the n-th figure D(n).

For n=1, the pattern is a single dot. D(1) = 1.
For n=2, the pattern is a 2x2 square, with 1 dot on top. D(2) = 2^2+1 = 4+1 = 5.
For n=3, the pattern is a 3x3 square, with (2+1) dots on top. D(3) = 3^2+(1+2) = 9+3 = 12.
For n=4, the pattern is a 4x4 square, with (3+2+1) dots on top. D(4) = 4^2+(1+2+3) = 16+6 = 22.

So there are two parts to the pattern. We always have an nxn square of dots; and above that we have a number of dots which is the sum of the whole numbers from 1 to (n-1).

You might be familiar with the formula for the sum of the integers from 1 to n; it is

So the sum fo the integers from 1 to (n-1) is

And so the number of dots in the n-th figure is

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