https://file.aiquickdraw.com/mathbot/file/172552918133555wx6hxo.jpg

The image shows a problem where a scientist observes bacteria growth and records the number of bacteria at regular intervals. The table in the image lists the observations as follows:

| Observation | 1 | 2 | 3 | 4 | 5 |
|-------------|---|---|---|---|---|
| Bacteria    | 3 | 9 | 27 | 81 | 243 |

The question asks how many bacteria will be present during the 12th observation.

### Solution:

From the table, the sequence of bacteria counts is 3, 9, 27, 81, 243. This sequence is a geometric sequence where each term is obtained by multiplying the previous term by a common ratio.

The common ratio \( r \) can be found as:
\[
r = \frac{9}{3} = \frac{27}{9} = \frac{81}{27} = \frac{243}{81} = 3
\]

The general formula for the \( n \)-th term of a geometric sequence is:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
where:
- \( a_1 = 3 \) (the first term),
- \( r = 3 \) (the common ratio),
- \( n = 12 \) (the observation number).

Substituting the values:
\[
a_{12} = 3 \cdot 3^{(12-1)} = 3 \cdot 3^{11}
\]

Let's calculate \( 3^{11} \).

\[
3^{11} = 177147
\]

Therefore:

\[
a_{12} = 3 \cdot 177147 = 531441
\]

So, the number of bacteria during the 12th observation will be **531,441**.

### Would you like more details or have any questions?

### Related Questions:
1. How do you find the common ratio in a geometric sequence?
2. What are the differences between arithmetic and geometric sequences?
3. How can you derive the general term formula for a geometric sequence?
4. How would you calculate the 20th term of this sequence?
5. What are real-life applications of geometric sequences in biology?

### Tip:
Always identify the pattern in sequences; it helps to easily predict future terms!

A scientist observes bacteria growth and records it in a table at regular intervals. How much bacteria will there be during the 12th observation?

Discover how to use geometric progression to predict bacteria growth. With a sequence starting at 3 bacteria and a common ratio of 3, learn how to calculate the 12th term, representing bacteria count. Perfect for students in grades 9-12.

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