Question

find the 5th term of the geometric sequence.
2, 6, 18,...
a 36 b 54 c 162 d 486

Explanation:

Step1: Identify the common ratio \( r \)

In a geometric sequence, the common ratio \( r \) is found by dividing a term by its previous term. For the sequence \( 2, 6, 18, \dots \), we calculate \( r=\frac{6}{2} = 3 \) (we can also check \( \frac{18}{6}=3 \), so the common ratio \( r = 3 \)).

Step2: Recall the formula for the \( n \)-th term of a geometric sequence

The formula for the \( n \)-th term of a geometric sequence is \( a_n=a_1\times r^{(n - 1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. Here, \( a_1 = 2 \), \( r=3 \), and we want to find the 5th term, so \( n = 5 \).

Step3: Substitute the values into the formula

Substitute \( a_1 = 2 \), \( r = 3 \), and \( n = 5 \) into the formula \( a_n=a_1\times r^{(n - 1)} \). We get \( a_5=2\times3^{(5 - 1)} \). First, calculate the exponent: \( 5-1 = 4 \). Then, calculate \( 3^4=81 \). Now, multiply by \( a_1 \): \( a_5=2\times81 = 162 \).

Answer:

C. 162