Question

Find a recursive formula for the sequence: 1, -1,-7,-25

Answer 1

Check the forward differences:

-1 - 1 = -2

-7 - (-1) = -6

-25 - (-7) = -18

Notice how the differences appear to follow a geometric progression with common ratio 3. So if $a_n$ denotes the $n$th term in the given sequence, we seem to have

$a_2-a_1=-2\cdot3^0$

$a_3-a_2=-2\cdot3^1$

$a_4-a_3=-2\cdot3^2$

so that the general pattern for $n>1$ would be

$a_n-a_{n-1}=-2\cdot3^{n-2}$

Then the sequence is given recursively by

$a_n=\begin{cases}1&\text{for }n=1\\a_{n-1}-2\cdot3^{n-2}&\text{for }n>1\end{cases}$

The first 10 terms in the sequence would be

1, -1, -7, -25, -79, -241, -727, -2185, -6559, -19681