How do you find the explicit formula for the following sequence 14, 21, 42, 77, ......?

1 Answer
Mar 11, 2016

Answer:

Examine sequences of differences to derive a formula:

#a_n = 7(n^2-2n+3)#

Explanation:

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.

#color(white)()#
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:

#color(blue)(14), 14#

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)#

#=14+7n-7+7n^2-21n+14#

#=7n^2-14n+21#

#=7(n^2-2n+3)#

where #n = 1, 2, 3,...#