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maria [59]
3 years ago
9

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

Mathematics
1 answer:
RoseWind [281]3 years ago
4 0

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 nth 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

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