Question

1. the first three terms of a geometric sequence are shown below. which of the following is the value of its 12th term? 4, 12, 36, ... (1) 236,196 (2) 708,588 (3) 2,125,764 (4) 12,582,912

Explanation:

Step1: Find the common ratio \( r \)

In a geometric sequence, the common ratio \( r \) is found by dividing a term by its previous term. So, \( r=\frac{12}{4} = 3 \) (or \( \frac{36}{12}=3 \)).

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, \( a_1 = 4 \), \( r = 3 \), and we want to find the 12th term, so \( n=12 \).

Step3: Substitute the values into the formula

Substitute \( a_1 = 4 \), \( r = 3 \), and \( n = 12 \) into the formula: \( a_{12}=4\times3^{12 - 1}=4\times3^{11} \).
First, calculate \( 3^{11}=177147 \). Then, \( 4\times177147 = 708588 \).

Answer:

(2) 708,588