To find the 25th term of the geometric sequence $2, -6, 18, \ldots$, we use the formula:

$a_n = a_1 \cdot r^{(n-1)}$

Where:

• $a_1 = 2$ (the first term),
• $r = \frac{-6}{2} = -3$ (the common ratio),
• $n = 25$ (the term number).

Now, substitute these values into the formula:

$a_{25} = 2 \cdot (-3)^{24}$

We know that $(-3)^{24}$ is a large positive number because raising a negative number to an even power results in a positive number.

Calculate $(-3)^{24}$:

$(-3)^{24} = 282429536481$

Now, multiply by 2:

$a_{25} = 2 \cdot 282429536481 = 564859072962$

Thus, the 25th term of the sequence is 564,859,072,962.

Would you like more details or have any questions? Here are some related questions to consider:

1. What is the 15th term of this geometric sequence?
2. What is the sum of the first 25 terms of the sequence?
3. How does the sequence change if the common ratio is changed to a positive number?
4. Can you find the general formula for the $n$th term of this sequence?
5. How does the sequence behave if we extend it to 50 terms?

Tip: When working with large exponents, consider properties of exponents (e.g., even or odd) to quickly determine the sign of the result.