Expert: Sherry Wallin - 6/3/2011

Question
"Hi Sherry

Question: if given a series of triangles composed of evenly spaced dots: .   .     .        etc. etc. determine the general statement       . .   . .
                    . . .
(x= the number of the triangle)(y= # of dots in the triangle)

I have already determined the statement as (x^2+x)/ 2. But I determined this using guess and check in a table. I was wondering how would I prove this formula using the diagrams of the triangles? I'm doing this as a Math30 project.

Thanks you so much for your help! Reply ASAp, if you could . Thanks again!

Answer
Can you put the actual question in a file and attach it? What you have written does not make sense. I don't see anywhere, where you have used y, unless you are saying y = (x^2+x)/2? You would then have to put a lower boundary on y because one dot will not make a triangle and neither will two dots...three dots is the minimum number of dots necessary to draw the smallest triangle. If you were to have one dot in the first row and then two dots centered under the first dot in the second row, that is when you get your first triangle. Then in the 3rd row you will have 3 dots centered under the 2nd row and that would give you the next smallest triangle. All you are doing is adding the consecutive integers from 1 to infinity for the number of dots. The sum of this series is [n(n+1)]/2 where you have used x for n. Every time you add another row you will get an additional triangle and you will be adding the next consecutive integer. So with two rows you get one triangle. With three rows you get two triangles, with four rows you get three triangles etc... notice the number of rows is always one more than the number of triangles and the number of dots is the sum of the row number. In the example above, when you have two rows you have three dots and one triangle. Let n = # of rows and n-1 = number of triangles, and 1 + 2 + ...+ n = number of dots.

Give me any row, say row 5, and we know that there are n = 5 rows, n-1 = 5-1 = 4 triangles, and 1 + 2 + 3 + 4 + 5 = 15 dots. You can think of this as an ordered triple (x,y,z) where x = number of rows, y = number of triangles, and z = number of dots or (x,y,z) = (n, n-1, 1 + 2 +...+ n).

Check it out for the first couple of values of n:
let n = 1, then you have 1 row, 0 triangles, and 1 dot
n = 2 , then you have 2 rows, i triangle, and 3 dots
n = 3, then you have 3 rows, 2 triangles, and 6 dots

and so on and so forth...

Math Prof