Question

find the $12^{\text{th}}$ term of the following geometric sequence.\
$5, 10, 20, 40, \dots$

Explanation:

Step1: Identify the first term and common ratio

The first term \( a_1 = 5 \). The common ratio \( r \) is found by dividing a term by its previous term, e.g., \( r=\frac{10}{5} = 2 \).

Step2: Use the geometric sequence formula

The formula for the \( n \)-th term of a geometric sequence is \( a_n=a_1\times r^{n - 1} \). For the 12th term, \( n = 12 \), \( a_1 = 5 \), \( r = 2 \). So \( a_{12}=5\times2^{12 - 1} \).

Step3: Calculate the exponent and then the term

First, \( 12-1 = 11 \), so \( a_{12}=5\times2^{11} \). Since \( 2^{11}=2048 \), then \( 5\times2048 = 10240 \).

Answer:

\( 10240 \)