Number patterns - finding the pattern The number pattern is 
$$1,3,6,10,15,\dots$$
Find formula for this pattern. And thus find the 50th pattern. 
I have problems in trying to come out with these formulas, is there a way to see patterns? Let's say if the pattern is changed, is the method still the same? I need advice on how to solve the above pattern as well as what is the method!
Thanks in advance !
 A: In these types of "puzzles" there are always several (in fact, infinitely many!) possible answers, but a simple one would be to notice that the difference between each successive term increases with one for each term, i.e.
$$a_{n+1}=a_n+(n+1), \quad \text{with }a_1=1.$$
This can be solved to give $$a_n=\frac{n(n+1)}{2},$$
which is the sum of the natural numbers up until and including $n$ (as Jan also noticed). 
These is no method in general, though there are a few things one should try first: Looking at the difference for the first few terms is one of them. Another would be to look for a pattern in all even terms (or odd terms). 
A: $$\sum_{m=1}^{n}m=\frac{n(n+1)}{2}$$
So, we get:
$$1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,\dots$$
A: I suggest you look up 'the method of finite differences' for general tips on solving these sorts of sequences.
A: The set $1, 3, 6, 10, 15, ...$ has a obvious pattern:
$ 1+(1+1)=3$
$ 3+(2+1) = 6$
$ 6+(3+1) = 10$
$ 10+(4+1) = 15$
This is the triangular number sequence.
See Triangular number. If you want to find the number of dots, flip a equilateral triangle so that it matches up with the first one. It forms a rectangle with width $ 1 + $triangle length and the height the same as the length of the triangle. The area of the rectangle is $n(n+1)$ where $n$ is the length of the triangle. That means that the number of dots (or the area) of the equilateral triangle is 
$$\frac{n(n+1)}{2}$$
The sequence of the number of dots in a triangle is the same as the sequence that you asked for. Easily plug in $n$ for $50$ and you get:
$$ \frac{50(50+1)}{2} = 1275$$ 
for the answer. It is the same way ($\frac{n(n+1)}{2}$)for any number greater than $1$.
