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

Mar 11, 2016

Examine sequences of differences to derive a formula:

${a}_{n} = 7 \left({n}^{2} - 2 n + 3\right)$

#### 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.

$\textcolor{w h i t e}{}$
Examine sequences of differences as follows...

Write out the original sequence:

$\textcolor{red}{14} , 21 , 42 , 77$

Write out the sequence of differences between pairs of adjacent terms:

$\textcolor{g r e e n}{7} , 21 , 35$

Write out the sequence of differences of those differences:

$\textcolor{b l u e}{14} , 14$

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)

$= 14 + 7 n - 7 + 7 {n}^{2} - 21 n + 14$

$= 7 {n}^{2} - 14 n + 21$

$= 7 \left({n}^{2} - 2 n + 3\right)$

where $n = 1 , 2 , 3 , \ldots$