The given sequence is:
$10, -20, 40, \dots$

This is a geometric sequence, where each term is obtained by multiplying the previous term by a common ratio $r$.

Step 1: Find the Common Ratio $r$

The common ratio is found using:
$r = \frac{\text{second term}}{\text{first term}} = \frac{-20}{10} = -2$ Thus, $r = -2$.

Step 2: Use the Geometric Sequence Formula

The general formula for the $n$th term of a geometric sequence is:
$a_n = a_1 \cdot r^{(n-1)}$ where:

• $a_1 = 10$ (first term),
• $r = -2$,
• $n = 10$.

Step 3: Compute the 10th Term

$a_{10} = 10 \cdot (-2)^{(10-1)}$ $= 10 \cdot (-2)^9$ $= 10 \cdot (-512)$ $= -5120$

Final Answer:

The 10th term of the sequence is $-5120$.

Would you like a step-by-step breakdown or further explanation?

Here are five related questions you might find interesting:

1. How do you determine whether a sequence is geometric or arithmetic?
2. What is the sum of the first 10 terms of this geometric sequence?
3. How does changing the common ratio affect the sequence's growth?
4. What happens if the common ratio is a fraction instead of an integer?
5. How do you find the general formula for any geometric sequence?

Tip: In a geometric sequence, if the common ratio is negative, the terms alternate in sign!