# What is the next term in the sequence 5, 4, 10, 35, 97 ?

Jan 27, 2017

Could be $220$, but really anything you like.

#### Explanation:

No finite initial sequence of numbers determines the following numbers, unless you are told that it is a particular kind of sequence.

You may be asked to identify a pattern, which may be reasonably apparent from the given sequence, but even that fails in the given example.

One method to help find patterns is to look at differences between successive terms, but that gives no clue with the current example:

$5 , 4 , 10 , 35 , 97$

$- 1 , 6 , 25 , 62$

$7 , 19 , 37$

$12 , 18$

$6$

The only constant sequence we arrive at is one with just one term. So all we can say is that if we match the given sequence with a quartic formula then we can find additional terms by duplicating the $6$ in the last line and reconstructing:

$5 , 4 , 10 , 35 , 97 , \textcolor{red}{220}$

$- 1 , 6 , 25 , 62 , \textcolor{red}{123}$

$7 , 19 , 37 , \textcolor{red}{61}$

$12 , 18 , \textcolor{red}{24}$

$6 , \textcolor{red}{6}$

$\textcolor{w h i t e}{}$
Footnote

I once tried this approach with the sequence:

$0 , 2 , 5 , 7 , 8 , 9 , 11$

getting the result $14$ for the next term.

The 'proper' answer was $100$.

I will let you figure out why.

$\textcolor{w h i t e}{}$
$\textcolor{w h i t e}{}$
$\textcolor{w h i t e}{}$
$\textcolor{w h i t e}{}$
$\textcolor{w h i t e}{}$
$\textcolor{w h i t e}{}$
$\textcolor{l i g h t g r e y}{\text{Clue: It was on a French website}}$