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Question Find the recursive rule for the sequence 4,−12,36,−108,…4, -12, 36, -108, \ldots.

Studdy Solution

STEP 1

Assumptions1. The sequence given is 4,−12,36,−108,…4,-12,36,-108, \ldots . The sequence is a geometric sequence, meaning each term is multiplied by a common ratio to get the next term.
3. The common ratio can be found by dividing any term by the previous term.

STEP 2

First, we need to find the common ratio of the sequence. We can do this by dividing any term by the previous term.
Common ratio=TermnTermn−1Common\, ratio = \frac{Term_{n}}{Term_{n-1}}

STEP 3

Now, plug in the given values for the second term and the first term to calculate the common ratio.
Common ratio=−12Common\, ratio = \frac{-12}{}

STEP 4

Calculate the common ratio.
Common ratio=−124=−3Common\, ratio = \frac{-12}{4} = -3

STEP 5

Now that we have the common ratio, we can write the recursive rule for the sequence. The recursive rule for a geometric sequence isTermn=Termn−1×Common ratioTerm_{n} = Term_{n-1} \times Common\, ratio

STEP 6

Plug in the values for the first term and the common ratio to write the recursive rule.
Termn=Termn−1×−3Term_{n} = Term_{n-1} \times -3

STEP 7

The first term of the sequence is4, so we can write the complete recursive rule asTerm1=4Term_{1} =4Termn=Termn−1×−3, for n>1Term_{n} = Term_{n-1} \times -3, \, for \, n >1So, the recursive rule for the sequence 4,−12,36,−108,…4,-12,36,-108, \ldots is Term1=4Term_{1} =4 and Termn=Termn−1×−3Term_{n} = Term_{n-1} \times -3 for n>1n >1.

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