Question

For the following geometric sequence find the recursive formula: {-1, 3, -9, ...}.

Answer 1

Based on these first few terms, we can deduce that the next term is computed by switching the sign of the previous one, and multiplying it by 3: we start with -1, we switch the sign (1) and multiply by 3 (3); then again we switch the sign (-3) and multiply by 3 (-9), and so on.

Since switching sign is the same as multiplying by -1, we can compute every next term by multiplying the previous one by -3:

$a_1 = -1\\a_2 = a_1\cdot(-3) = (-1)\cdot(-3)=3\\a_3 =a_2\cdot(-3)=3\cdot(-3)=-9$

So, the recursive formula is

$a_n = -3a_{n-1}$

because it states precisely that the next term is -3 times the previous one.