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For a geometric sequence, the following relationship between the first and third terms holds:

a₁ − a₃ = −1.5
a₃ − a₁ = 1.5

Calculate the common ratio q and the first term a₁.

Final Answer:

q =  2
a₁ =  0.5

Step-by-step explanation:

$$\begin{gathered} a_1-a_3=-1.5 \\ {a}_3-a_1=1.5 \\ {a}_2=q{a}_1 \\ {a}_3=q{a}_2=q^2{a}_1 \\ {a}_1-q^2{a}_1=-1.5 \\ {q}^2{a}_1-a_1=1.5 \\ {a}_1(1-q^2)=-1.5 \\ {a}_1(q^2-1)=1.5 \\ q=2 \\ {a}_1=\frac{1.5}{q^2-1}=\frac{1.5}{2^2-1}=\frac{1}{2}=0.5 \end{gathered}$$

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