Number patterns - finding the pattern

Question

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 !

Answer 1

$$\sum_{m=1}^{n}m=\frac{n(n+1)}{2}$$

So, we get:

$$1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,\dots$$

Answer 2

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).

Answer 3

Let $a_n$ be the number of ordered triples $(x,y,z)$ of palindromes such that $x+y+z=n$, then the first five items of $a_n$ match the first five numbers you gave.

You can find it in oeis and N. J. A. Sloane provided a maple program to calculate it. In this way $a_{50}=36$.

Answer 4

Let me give you the basic starting points for finding a pattern:

Calculate the subsequent fractions:

1, 3, 6, 10, 15, ... => $\frac{3}{1}=3$, $\frac{6}{3}=2$, $\frac{10}{6}=1.666...$, $\frac{15}{10}=1.5, ...$

=> Although it starts promising, it leads to nothing!

Calculate the subsequent subtractions:

1, 3, 6, 10, 15, ... => $3 - 1=2, 6 - 3=3, 10 - 6=4, 15 - 10=5$

=> a series of 1, 2, 3, 4, 5 clearly is the way to go.