Triangular numbers

Question

is there a triangular number that has 91 dots in its shape? if so, which one is it? in other words, where does it appear in the sequence of triangular numbers? is it the 5th, the 12th, etc.? is there a triangular number that has 150 dots in its shape? what about 2000 dots? if so, where are these numbers in the sequence of triangular numbers? if there are no numbers with 150 or 2000 dots, how do you know for sure?

Answer 1

1 dot in top row; following rows have these numbers of dots:

2

3

4

5

6

7

8...

 

The sum of all these numbers obeys a rule n(n+1)/2 so the associated triangular numbers are 

1,3,6,10,15,21,...where n is the row number. The formula is the well known one for Sum of first n positive integers.

 

If we set the above expression = 91, we have the equation n(n+1) = 182, or n²+n-182 = 0.

Factoring, (n+14)(n-13) = 0  so n=13 (this is the ROW # that you want).

 

With this formula & method, you should be able to answer many question about Triangular Numbers.