Find the 12th term of the geometric sequence 10, comma, minus, 50, comma, 250, comma, point, point, .10,−50,250,...

To find the 12th term of the geometric sequence \(10, -50, 250, \ldots\), we start by identifying the first term (\(a\)) and the common ratio (\(r\)).

1. **First term \(a\)**: The first term \(a = 10\).
2. **Common ratio \(r\)**: To find the common ratio, divide the second term by the first term:
   \[
   r = \frac{-50}{10} = -5
   \]

The formula for the \(n\)th term of a geometric sequence is:
\[
a_n = a \times r^{n-1}
\]

To find the 12th term (\(a_{12}\)):
\[
a_{12} = 10 \times (-5)^{12-1}
\]
\[
a_{12} = 10 \times (-5)^{11}
\]

Now calculate \((-5)^{11}\):
\[
(-5)^{11} = -48828125
\]

So:
\[
a_{12} = 10 \times (-48828125) = -488281250
\]

Therefore, the 12th term is \(-488281250\).

Would you like a detailed breakdown of these steps or do you have any questions?

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Here are 5 related questions you might find useful:

1. How do you find the sum of the first \(n\) terms in a geometric series?
2. What is the formula for the sum of an infinite geometric series?
3. How do you determine if a series is convergent or divergent?
4. Can a geometric sequence have a common ratio that is a fraction? What would that look like?
5. How do you find the geometric mean between two numbers?

**Tip:** Always check the sign of the common ratio carefully, as it determines the alternating pattern in the sequence!

Learn how to find the 12th term of the geometric sequence 10, -50, 250 with a step-by-step solution. This problem covers geometric sequences, common ratio calculation, and nth term formula application. Suitable for high school students.

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