find the 10th term of the geometric sequence 10, -20, 40, ...

Explanation:

Step1: Find the common ratio

The common ratio $r$ of a geometric sequence is found by dividing a term by its previous term. So $r=\frac{-20}{10}=- 2$.

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, $a_1 = 10$, $r=-2$, and $n = 10$. Substitute these values into the formula: $a_{10}=10\times(-2)^{10 - 1}$.

Step3: Calculate the value of $(-2)^9$

$(-2)^9=(-1)^9\times2^9=-512$.

Step4: Calculate the 10th term

$a_{10}=10\times(-512)=-5120$.

Answer:

$-5120$