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Math  /  Algebra

QuestionWhat are the first three terms of the geometric sequence an=−7⋅3n−1a_{n}=-7 \cdot 3^{n-1} ? a1=Ex⁡1232^a2=Ex⁡1232^a3=Ex⋅1232^\begin{array}{l} a_{1}=\operatorname{Ex} 123 \hat{2} \\ a_{2}=\operatorname{Ex} 123 \hat{2} \\ a_{3}=E x \cdot 123 \hat{2} \end{array}

Studdy Solution

STEP 1

1. The sequence given is a geometric sequence.
2. The general term of the sequence is given by an=−7⋅3n−1 a_{n} = -7 \cdot 3^{n-1} .
3. We need to find the first three terms of this sequence.

STEP 2

1. Calculate the first term a1 a_1 .
2. Calculate the second term a2 a_2 .
3. Calculate the third term a3 a_3 .

STEP 3

To find the first term a1 a_1 , substitute n=1 n = 1 into the formula for the sequence:
a1=−7⋅31−1 a_1 = -7 \cdot 3^{1-1} a1=−7⋅30 a_1 = -7 \cdot 3^0 a1=−7⋅1 a_1 = -7 \cdot 1 a1=−7 a_1 = -7

STEP 4

To find the second term a2 a_2 , substitute n=2 n = 2 into the formula for the sequence:
a2=−7⋅32−1 a_2 = -7 \cdot 3^{2-1} a2=−7⋅31 a_2 = -7 \cdot 3^1 a2=−7⋅3 a_2 = -7 \cdot 3 a2=−21 a_2 = -21

STEP 5

To find the third term a3 a_3 , substitute n=3 n = 3 into the formula for the sequence:
a3=−7⋅33−1 a_3 = -7 \cdot 3^{3-1} a3=−7⋅32 a_3 = -7 \cdot 3^2 a3=−7⋅9 a_3 = -7 \cdot 9 a3=−63 a_3 = -63
The first three terms of the geometric sequence are:
a1=−7,a2=−21,a3=−63 a_1 = -7, \quad a_2 = -21, \quad a_3 = -63

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