How do you find the explicit formula for the following sequence 14, 21, 42, 77, ......?
Examine sequences of differences to derive a formula:
#a_n = 7(n^2-2n+3)#
Disclaimer: A finite sequence does not determine a unique formula for the terms of the sequence. The steps below assume that once we find a sequence that appears to be constant, then it will carry on being constant.
Examine sequences of differences as follows...
Write out the original sequence:
#color(red)(14), 21, 42, 77#
Write out the sequence of differences between pairs of adjacent terms:
#color(green)(7), 21, 35#
Write out the sequence of differences of those differences:
Having reached a constant sequence, we can use the initial terms of each of these sequences as coefficients in a formula for a general term of the sequence:
#a_n = color(red)(14)/(0!) + color(green)(7)/(1!)(n-1) + color(blue)(14)/(2!)(n-1)(n-2)#