find the 25th term of the geometric sequence 2, -6, 18,

To find the 25th term of a geometric sequence, we use the formula:

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

Where:
- \( a_n \) is the \( n \)-th term of the sequence.
- \( a_1 \) is the first term of the sequence.
- \( r \) is the common ratio.
- \( n \) is the term number.

Given:
- \( a_1 = 2 \)
- The common ratio \( r \) can be found by dividing the second term by the first term: \( r = \frac{-6}{2} = -3 \)
- \( n = 25 \)

Now, substitute the values into the formula:

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

Now, let's calculate this step by step.

1. Calculate \( (-3)^{24} \).
2. Multiply the result by 2.

\( (-3)^{24} \) results in a very large positive number because raising a negative number to an even exponent gives a positive result.

\[
(-3)^{24} = 282429536481
\]

Now, multiply by 2:

\[
a_{25} = 2 \cdot 282429536481 = 564859072962
\]

So, the 25th term of the geometric 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 10th term of the same sequence?
2. How would the sequence change if the common ratio were positive?
3. Can you find the sum of the first 25 terms of this geometric sequence?
4. What is the significance of the sign of the common ratio in a geometric sequence?
5. How does the sequence behave as \( n \) approaches infinity?

**Tip:** In geometric sequences, the sign and magnitude of the common ratio determine whether the sequence converges, diverges, or alternates in sign.

Learn how to find the 25th term of the geometric sequence 2, -6, 18 using the formula for the nth term of a geometric sequence. Detailed step-by-step solution provided.

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