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:

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!