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QuestionFind the explicit formula for the sequence −7,−4,−7,2,5-7, -4, -7, 2, 5. Choose from A, B, C, or D.

Studdy Solution

STEP 1

Assumptions1. The sequence given is −7,−4,−7,,5-7,-4,-7,,5 . We are looking for an explicit formula of the form an=a+(n−1)da_{n}=a+(n-1)d, where aa is the first term and dd is the common difference3. The sequence is not necessarily arithmetic, but the formula options suggest it might be

STEP 2

First, let's try to identify if there's a common difference between consecutive terms. This would indicate an arithmetic sequence.
d=an−an−1d = a_{n} - a_{n-1}

STEP 3

Calculate the differences between consecutive terms.
d1=−−(−7)=3d1 = - - (-7) =3d2=−7−(−)=−3d2 = -7 - (-) = -3d3=2−(−7)=9d3 =2 - (-7) =9d=5−2=3d =5 -2 =3

STEP 4

We can see that the differences are not constant, so the sequence is not arithmetic. However, we can observe a pattern the differences alternate between3 and -3, starting with3.

STEP 5

Let's try to formulate an explicit formula for the sequence based on this pattern. We can start with the first term, a1=−7a_{1} = -7, and then add the alternating differences.

STEP 6

We can express the alternating differences as (−1)n+1⋅3(-1)^{n+1} \cdot3. This will give us3 for odd nn and -3 for even nn.

STEP 7

Now, we can write the explicit formula asan=a1+(n−1)⋅((−1)n+1⋅3)a_{n} = a_{1} + (n-1) \cdot ((-1)^{n+1} \cdot3)

STEP 8

implify the formula to match the given optionsan=−7+(n−1)⋅((−1)n+1⋅3)a_{n} = -7 + (n-1) \cdot ((-1)^{n+1} \cdot3)

STEP 9

Comparing this with the given options, we see that it matches with option B.
So, the explicit formula for the sequence is an=−7+(n−)3a_{n}=-7+(n-)3.

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