Question

find the $12^{\text{th}}$ term of the following geometric sequence.\
$2, 8, 32, 128, \dots$

Explanation:

Step1: Identify the first term and common ratio

In a geometric sequence, the first term \(a_1 = 2\). The common ratio \(r\) is found by dividing a term by its previous term. For example, \(r=\frac{8}{2} = 4\), or \(\frac{32}{8}=4\), so \(r = 4\).

Step2: Use the geometric sequence formula

The formula for the \(n\)-th term of a geometric sequence is \(a_n=a_1\times r^{n - 1}\). We want to find the 12th term, so \(n = 12\), \(a_1=2\), \(r = 4\).
Substitute these values into the formula: \(a_{12}=2\times4^{12 - 1}\)

Step3: Simplify the expression

First, calculate the exponent: \(12-1 = 11\), so we have \(a_{12}=2\times4^{11}\).
We know that \(4^{11}=(2^2)^{11}=2^{22}\), so \(a_{12}=2\times2^{22}=2^{23}\).
Calculate \(2^{23}\): \(2^{10}=1024\), \(2^{20}=(2^{10})^2 = 1024^2=1048576\), \(2^{23}=2^{20}\times2^{3}=1048576\times8 = 8388608\).

Answer:

\(8388608\)