# How do you find the nth term of the sequence 1, 3, 6, 10, 15,...?

Feb 8, 2017

${a}_{n} = \frac{1}{2} n \left(n + 1\right)$

#### Explanation:

These are recognisable as triangular numbers, but let's use a general method for finding matching polynomial formulas...

Write down the initial sequence:

$\textcolor{red}{1} , 3 , 6 , 10 , 15$

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

$\textcolor{m a \ge n t a}{2} , 3 , 4 , 5$

Write down the sequence of differences of those differences:

$\textcolor{b l u e}{1} , 1 , 1$

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 = color(red)(1)/(0!) + color(magenta)(2)/(1!)(n-1) + color(blue)(1)/(2!)(n-1)(n-2)

$\textcolor{w h i t e}{{a}_{n}} = 1 + 2 n - 2 + \frac{1}{2} {n}^{2} - \frac{3}{2} n + 1$

$\textcolor{w h i t e}{{a}_{n}} = \frac{1}{2} {n}^{2} + \frac{1}{2} n$

$\textcolor{w h i t e}{{a}_{n}} = \frac{1}{2} n \left(n + 1\right)$