Question

find the 13th term of the following geometric sequence.
4, 12, 36, 108, ...

Explanation:

Step1: Identify the geometric sequence parameters

In a geometric sequence, the first term \(a_1 = 4\), and the common ratio \(r\) is found by dividing a term by its previous term. For example, \(r=\frac{12}{4} = 3\).

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 need to find the 13th term, so \(n = 13\), \(a_1=4\), \(r = 3\).
Substitute the values into the formula: \(a_{13}=4\times3^{13 - 1}\)

Step3: Calculate the exponent

First, simplify the exponent: \(13-1 = 12\), so \(a_{13}=4\times3^{12}\)

Step4: Calculate \(3^{12}\)

\(3^{12}=531441\)

Step5: Multiply by 4

\(a_{13}=4\times531441 = 2125764\)

Answer:

\(2125764\)