To get a human or AI tutor to help you, click Register
Sample 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.
The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term?
\(\text {(A) } -\sqrt{3} \qquad \text {(B) } \frac{-2\sqrt{3}}{3} \qquad \text {(C) } \frac{-\sqrt{3}}{3} \qquad \text {(D) } \sqrt{3} \qquad \text {(E) } 3\)
Answer Keys
Question 6: BSolutions
Question 6Step 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.