How to find the nth term of the sequence 1 , 3 , 6 , 10 , 15 , ... ?

Question

How to find the nth term of the sequence       1    ,    3    ,    6    ,    10    ,    15    ,    ...  ?

Nathan Franco · Accepted Answer

These are clearly triangular numbers, but let's find matching polynomial formulas using a more general approach.
Write down the initial sequence:

1, 3, 6, 10, 15

Write down the sequence of differences between consecutive pairs of terms:

2, 3, 4, 5

Write down the sequence of differences of those differences:

1, 1, 1

Having reached a constant sequence, we can write down a formula for the

n

th term using the initial term of each of these sequences as a coefficient:

a_{n} = \frac{1}{0!} + \frac{2}{1!} (n - 1) + \frac{1}{2!} (n - 1) (n - 2)

a_{n} = 1 + 2 n - 2 + \frac{1}{2} n^{2} - \frac{3}{2} n + 1

a_{n} = \frac{1}{2} n^{2} + \frac{1}{2} n

a_{n} = \frac{1}{2} n (n + 1)

How to find the nth term of the sequence 1, 3, 6, 10, 15,...?