Finding expression for pattern

Question

This seems easy....

Pattern:

3, 6, 10, 15, 21....

How do I find expression for this?

It looked easy, but I couldn't find it.

I tried graphing it, but it turned out that it is not quadratic...

Thanks.

Answer 1

$f(n)=\binom{n+2}{2}=\frac{(n+1)(n+2)}{2}$

If you have a sequence where the differences between terms is lineal then it is going to be a quadratic polynomial.

Answer 2

If you don't recognize the pattern (Triangular numbers), you can look at the differences.

They are 3,4,5,6. Since the differences are an arithmetic sequence, you know the series is a quadratic, so can start with $n^2$.

Then, you can call your expression $an^2+bn+c$ and solve it by using 3 of the values from your sequence.

Answer 3

$$3\quad 3+\color{red}3=6\quad 6+\color{red}4=10\quad 10+\color{red}5=15\quad 15+\color{red}6=21\quad $$

Answer 4

$$~~~~~~3 ~~~~~~ 6 ~~~~~~ 10 ~~~~~~ 15 ~~~~~~ 21 \\3 ~~~~~~~~ 4 ~~~~~~~~~ 5 ~~~~~~ 6 \\~1 ~~~~~~~~ 1 ~~~~~~~ 1$$

So $f(x) = ax^2 + bx + c$ Now $f(1)=3,f(2)=6,f(3)=10.$ From this we can solve $a,b,c$

Answer 5

If you prefer recursion: $a_n=a_{n-1}+n-1+3$ with $a_1=3$.