QuestionFind the explicit formula for the geometric sequence: $-6,-18,-54,-162,\ldots$ . Options: A, B, C, D.

Studdy Solution

STEP 1

Assumptions1. The sequence is geometric, meaning each term is the product of the previous term and a constant ratio. . We need to find the explicit formula for the nth term of the sequence.

STEP 2

In a geometric sequence, the nth term is given by the formula $a_{n} = a_{1} \times r^{(n-1)}$ where- $a_{n}$ is the nth term, - $a_{1}$ is the first term, - $r$ is the common ratio, - $n$ is the term number.

STEP 3

Identify the first term ( $a_{1}$ ) and the common ratio ( $r$ ) from the given sequence.
The first term $a_{1}$ is -6.
The common ratio $r$ can be found by dividing any term by its preceding term. For instance, divide the second term by the first term $r = \frac{-18}{-6} =3$

STEP 4

Substitute the values of $a_{1}$ and $r$ into the formula for the nth term of a geometric sequence.
$a_{n} = -6 \times3^{(n-1)}$

STEP 5

Compare this formula with the given options.
The formula we derived matches option B.
Therefore, the explicit formula for the given geometric sequence is $a_{n}=-(3)^{(n-1)}$ .

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