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The question asks for the size of a population of insects after 10 months, given that the population starts with 5 members and triples every month.

### Step-by-Step Solution:

1. **Identify the Initial Value and Growth Factor:**
   - Initial population, \( P_0 = 5 \).
   - The population triples every month, meaning the growth factor is \( 3 \).

2. **Formulate the Exponential Growth Equation:**
   - The population after \( n \) months can be expressed as:
   \[
   P(n) = P_0 \times 3^n
   \]
   where \( P_0 = 5 \) and \( n \) is the number of months.

3. **Calculate the Population After 10 Months:**
   \[
   P(10) = 5 \times 3^{10}
   \]

4. **Compute the Value:**
   Let's compute the exact value.

\[
P(10) = 5 \times 3^{10} = 5 \times 59049 = 295245
\]

So, the population size after 10 months is approximately **295,245**.

### Answer:
The size of the population after 10 months is **295,245**.

Would you like more details on this solution or have any further questions?

### Related Questions:
1. What is the population after 5 months?
2. How would the formula change if the population quadrupled every month?
3. What if the initial population was 10 instead of 5?
4. Can this problem be modeled with a continuous growth formula instead of a discrete one?
5. What would be the population after 15 months?

**Tip:** Exponential growth can lead to very large numbers quickly, which is important in fields like biology and finance!

A population of insects starts with 5 members in the first generation and triples every month. Calculate, to the nearest integer, the size of the population after 10 months.

Learn how to calculate the size of an insect population after 10 months with exponential growth. Starting with 5 members and tripling each month, this problem explores the exponential growth formula P(n) = P_0 * 3^n, suitable for students in Grades 7-10.

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