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
denotes the
th term in the given sequence, we seem to have
![a_2-a_1=-2\cdot3^0](https://tex.z-dn.net/?f=a_2-a_1%3D-2%5Ccdot3%5E0)
![a_3-a_2=-2\cdot3^1](https://tex.z-dn.net/?f=a_3-a_2%3D-2%5Ccdot3%5E1)
![a_4-a_3=-2\cdot3^2](https://tex.z-dn.net/?f=a_4-a_3%3D-2%5Ccdot3%5E2)
so that the general pattern for
would be
![a_n-a_{n-1}=-2\cdot3^{n-2}](https://tex.z-dn.net/?f=a_n-a_%7Bn-1%7D%3D-2%5Ccdot3%5E%7Bn-2%7D)
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}](https://tex.z-dn.net/?f=a_n%3D%5Cbegin%7Bcases%7D1%26%5Ctext%7Bfor%20%7Dn%3D1%5C%5Ca_%7Bn-1%7D-2%5Ccdot3%5E%7Bn-2%7D%26%5Ctext%7Bfor%20%7Dn%3E1%5Cend%7Bcases%7D)
The first 10 terms in the sequence would be
1, -1, -7, -25, -79, -241, -727, -2185, -6559, -19681