Question

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

Answer 1

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,...`