# What is the formula for the sequence 2, -1, 4, -7, 10, -13, 16,... ?

Aug 20, 2017

The given sequence is matched by the formula:

${a}_{n} = {\left(- 1\right)}^{n} \left(- 2 + 3 \left(n - 1\right)\right)$

or if you prefer:

${a}_{n} = {\left(- 1\right)}^{n} \left(3 n - 5\right)$

#### Explanation:

No infinite sequence is determined purely by a finite number of terms, unless you are given further information - e.g. that the sequence is arithmetic or geometric.

That having been said, note that if we multiply the terms of the given sequence by ${\left(- 1\right)}^{n}$, then we get the sequence:

$- 2 , 1 , 4 , 7 , 10 , 13 , 16 , \ldots$

which is (as far as it goes) an arithmetic sequence with initial term $- 2$ and common difference $3$.

So a formula that fits the original sequence can be written:

${a}_{n} = {\left(- 1\right)}^{n} \left(- 2 + 3 \left(n - 1\right)\right)$