Sample Question 6:

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

$\text {(A) } -\sqrt{3} \qquad \text {(B) } \frac{-2\sqrt{3}}{3} \qquad \text {(C) } \frac{-\sqrt{3}}{3} \qquad \text {(D) } \sqrt{3} \qquad \text {(E) } 3$

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Answer Keys

Question 6: B

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Solutions

Question 6
Step 1: Define the first term of your geometric sequence as $a$ and the common ratio as $r$.

Step 2: Write the equations of the second and fourth terms, which are $ar = 2$ and $ar^3 = 6$ respectively.

Step 3: To eliminate $a$, divide the second equation by the first one. This will give you $r^2 = 3$.

Step 4: Now solve the equation for $r$, giving you $r = \pm\sqrt{3}$.

Step 5: Substitute $r$ back into the first equation $ar = 2$, to get $a = \frac{2}{\pm\sqrt{3}}=\pm\frac{2\sqrt{3}}{3}$.

Step 6: Thus, the possible value of the first term, $a$, could be $-\frac{2\sqrt{3}}{3}$, hence option B is the correct answer.