Answered

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]



Answer :

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]