Question

which are the first five terms of a geometric sequence in which the second term is 6 and the fourth term is 54?\
3, 6, 27, 54, 108\
-18, 6, 30, 54, 78\
\\(\frac{2}{3}\\), 6, 54, 486, 4,374\
2, 6, 18, 54, 162

Explanation:

Step1: Recall Geometric Sequence Formula

A geometric sequence has the form \( a_n = a_1 \cdot r^{n - 1} \), where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. Given the second term \( a_2 = 6 \), so \( a_2 = a_1 \cdot r = 6 \). We need to find which sequence has \( a_4 = 54 \). Using the formula, \( a_4 = a_1 \cdot r^{3}=54 \). Since \( a_1 \cdot r = 6 \), we can substitute \( a_1=\frac{6}{r} \) into \( a_4 \) formula: \( \frac{6}{r}\cdot r^{3}=54 \), which simplifies to \( 6r^{2}=54 \), then \( r^{2}=9 \), so \( r = \pm3 \). Now check each option:

Step2: Check Option 1 (3, 6, 27, 54, 108)

Common ratio \( r=\frac{6}{3}=2 \), but \( r^{2}=4
eq9 \), so not valid.

Step3: Check Option 2 (-18, 6, 30, 54, 78)

Common ratio \( r=\frac{6}{-18}=-\frac{1}{3} \), \( r^{2}=\frac{1}{9}
eq9 \), invalid.

Step4: Check Option 3 (\(\frac{2}{3}\), 6, 54, 486, 4374)

Common ratio \( r=\frac{6}{\frac{2}{3}} = 9 \), \( r^{2}=81
eq9 \), invalid.

Step5: Check Option 4 (2, 6, 18, 54, 162)

Common ratio \( r=\frac{6}{2}=3 \), \( r^{2}=9 \), which matches. Now check \( a_4 \): \( a_4 = a_1\cdot r^{3}=2\cdot3^{3}=2\cdot27 = 54 \), which is correct.

Answer:

The sequence is 2, 6, 18, 54, 162 (the fourth option in the list).