What is the formula to this math sequence: 1, 3, 7, 14?
1 Answer
It could be
Explanation:
You can always find a polynomial that matches a finite sequence like this one, but there are infinitely many possibilities.
Write out the original sequence:
#color(blue)(1),3,7,14#
Write out the sequence of differences:
#color(blue)(2),4,7#
Write out the sequence of differences of those differences:
#color(blue)(2),3#
Write out the sequence of differences of those differences:
#color(blue)(1)#
Having reached a constant sequence (!), we can write out a formula for
#a_n = color(blue)(1)/(0!)+color(blue)(2)/(1!)(n-1)+color(blue)(2)/(2!)(n-1)(n-2)+color(blue)(1)/(3!)(n-1)(n-2)(n-3)#
#=color(red)(cancel(color(black)(1)))+2n-color(red)(cancel(color(black)(2)))+color(red)(cancel(color(black)(n^2)))-3n+color(red)(cancel(color(black)(2)))+1/6n^3-color(red)(cancel(color(black)(n^2)))+11/6n-color(red)(cancel(color(black)(1)))#
#=(n^3+5n)/6#