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The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term? Type the letter of the correct option.

A. $$-\sqrt{3}$$

B. $$-\frac{2\sqrt{3}}{3}$$

C. $$-\frac{\sqrt{3}}{3}$$

D. $$\sqrt{3}$$
E. 3

Jun 30, 2020

Let the first term be $a$ and the common difference be $r$. Therefore, $ar=2\ \ (1) \qquad \text{and} \qquad ar^3=6\ \ (2)$ Dividing $(2)$ by $(1)$ eliminates the $a$, yielding $r^2=3$, so $r=\pm\sqrt{3}$. Now, since $ar=2$, $a=\frac{2}{r}$, so $a=\frac{2}{\pm\sqrt{3}}=\pm\frac{2\sqrt{3}}{3}$. We therefore see that $\boxed{\textbf{(B)}\ -\frac{2\sqrt{3}}{3}}$ is a possible first term.