solve the following using geometric sequences:
a) a population of ants is growing at a rate of 8% a year. if there are 160 ants in the initial population, hence find the number of ants after 6 years.

In this problem, we are dealing with a geometric sequence because the population of ants grows by a fixed percentage each year. Let's break down the solution step by step:

### Formula for a Geometric Sequence:
The general formula for the nth term of a geometric sequence is:

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

Where:
- \( a_n \) is the population after \( n \) years,
- \( a_1 \) is the initial population,
- \( r \) is the common ratio (growth factor),
- \( n \) is the number of years.

### Step 1: Identify the known values
- Initial population \( a_1 = 160 \) ants,
- Growth rate is 8%, so the growth factor \( r = 1 + \frac{8}{100} = 1.08 \),
- We are asked to find the population after 6 years, so \( n = 6 \).

### Step 2: Apply the formula
Now substitute the known values into the geometric sequence formula:

\[
a_6 = 160 \times 1.08^{6}
\]

### Step 3: Calculate the population
We will now calculate \( a_6 \).

\[
a_6 = 160 \times 1.08^6
\]

Let me calculate that.The population of ants after 6 years will be approximately 254 ants.

Would you like any more details or have any questions? Here are 5 related questions to expand on this topic:

1. How would the population change if the growth rate was 5% instead of 8%?
2. What is the population after 10 years instead of 6 years?
3. If the population decreased by 3% per year, what would it be after 6 years?
4. How long would it take for the population to double with the same 8% growth rate?
5. How can the geometric sequence formula be applied to a scenario with monthly growth?

**Tip**: When working with percentages, always convert the percentage to a decimal by dividing by 100 before using it in calculations.

Solve the following using geometric sequences: a) A population of ants is growing at a rate of 8% a year. If there are 160 ants in the initial population, hence find the number of ants after 6 years.

Explore how to calculate the future population of ants growing at 8% annually using geometric sequences. With an initial population of 160 ants, this example demonstrates how exponential growth affects population over 6 years. Suitable for Grades 9-12 students.

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