What is the formula to this math sequence: 1, 3, 7, 14?

1 Answer
Nov 3, 2015

Answer:

It could be #a_n = (n^3+5n)/6#

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# using the first element of each sequence as a coefficient:

#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#