find the 13th term of the geometric sequence 4, -16, 64, ...

Explanation:

Step1: Find the common ratio

The common ratio $r$ of a geometric - sequence is found by dividing a term by its previous term. For the sequence $4,-16,64,\cdots$, $r=\frac{-16}{4}=-4$.

Step2: Use the formula for the $n$th term of a geometric sequence

The formula for the $n$th term of a geometric sequence is $a_n=a_1r^{n - 1}$, where $a_1$ is the first - term, $r$ is the common ratio, and $n$ is the term number. Here, $a_1 = 4$, $r=-4$, and $n = 13$. Substitute these values into the formula: $a_{13}=4\times(-4)^{13 - 1}$.

Step3: Simplify the expression

First, calculate the exponent: $13−1 = 12$. Then, $(-4)^{12}=4^{12}$ (since the exponent is even). So, $a_{13}=4\times4^{12}$. Using the rule of exponents $a^m\times a^n=a^{m + n}$, we have $a_{13}=4^{1+12}=4^{13}$. $4^{13}=268435456$.

Answer:

$268435456$