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Select the correct answer.
What is the [tex]$n$[/tex]th term of the geometric sequence that has a common ratio of 6 and 24 as its third term?
A. [tex]$a_n = 24(6)^{n-1}$[/tex] B. [tex]$a_n = \frac{2}{3}(6)^{n-1}$[/tex] C. [tex]$a_n = 24(6)^n$[/tex] D. [tex]$a_n = \frac{3}{2}(6)^{n-1}$[/tex]
To determine the correct form of the [tex]$n$[/tex]th term of the geometric sequence, let's follow these steps:
1. Identify the formula for the [tex]$n$[/tex]th term of a geometric sequence: [tex]\[
a_n = a \cdot r^{n-1}
\][/tex] where [tex]\( a \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio.
2. Identify the given values: - The common ratio ([tex]\( r \)[/tex]) is 6. - The third term ([tex]\( a_3 \)[/tex]) is 24.
3. Express the third term in terms of the first term and common ratio: [tex]\[
a_3 = a \cdot r^2
\][/tex]
4. Substitute the known values into this equation: [tex]\[
24 = a \cdot 6^2
\][/tex]
5. Solve for the first term [tex]\( a \)[/tex]: [tex]\[
24 = a \cdot 36 \implies a = \frac{24}{36} = \frac{2}{3}
\][/tex]
6. Substitute [tex]\( a \)[/tex] and [tex]\( r \)[/tex] back into the [tex]$n$[/tex]th term formula: [tex]\[
a_n = \frac{2}{3} \cdot 6^{n-1}
\][/tex]
Given the options, you need to find the one that matches this formula:
A. [tex]\( a_n = 24(6)^{n-1} \)[/tex]
B. [tex]\( a_n = \frac{2}{3}(6)^{n-1} \)[/tex]
C. [tex]\( a_n = 24(6)^n \)[/tex]
D. [tex]\( a_n = \frac{3}{2}(6)^{n-1} \)[/tex]
The correct answer is: [tex]\[ \boxed{\frac{2}{3}(6)^{n-1}} \][/tex]
Thus, the correct choice is: [tex]\[ \boxed{B} \][/tex]
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