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)#