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

##### 1 Answer

Feb 8, 2017

#### Explanation:

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

Write down the initial sequence:

#color(red)(1), 3, 6, 10, 15#

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

#color(magenta)(2), 3, 4, 5#

Write down the sequence of differences of those differences:

#color(blue)(1), 1, 1#

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

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

#color(white)(a_n) = 1+2n-2+1/2n^2-3/2n+1#

#color(white)(a_n) = 1/2n^2+1/2n#

#color(white)(a_n) = 1/2n(n+1)#