We are given the population data for a town that seems to follow an arithmetic sequence:

• Population in 2010: 5000
• Population for the next five successive years: 5421, 5836, 6293, 6725, 7202

Step 1: Verify Arithmetic Sequence

In an arithmetic sequence, the difference between consecutive terms (called the common difference, $d$) remains constant. We can check this by calculating the differences between the successive populations:

• $5421 - 5000 = 421$
• $5836 - 5421 = 415$
• $6293 - 5836 = 457$
• $6725 - 6293 = 432$
• $7202 - 6725 = 477$

While the differences are not exactly the same, they are close enough, suggesting an approximate arithmetic sequence.

Step 2: Calculate the Average Common Difference

Since the differences vary slightly, we will take the average of these differences to find an approximate common difference:

$d = \frac{421 + 415 + 457 + 432 + 477}{5} = \frac{2202}{5} = 440.4$

Thus, the approximate common difference is $d = 440.4$.

Step 3: Predict Population in 2024

The year 2024 is 14 years after 2010, so we need to find the population after 14 terms. Using the formula for the $n$-th term of an arithmetic sequence:

$a_n = a_1 + (n-1)d$

Where:

• $a_1 = 5000$ (the population in 2010),
• $d = 440.4$ (the common difference),
• $n = 15$ (since 2024 is the 15th year after 2010).

Now, we calculate $a_{15}$:

$a_{15} = 5000 + (15-1) \times 440.4 = 5000 + 14 \times 440.4 = 5000 + 6165.6 = 11165.6$

So, the predicted population in 2024 is approximately 11,200 (to three significant figures).

Summary:

• Common difference: $440.4$
• Predicted population in 2024: $11,200$

Do you have any questions, or would you like more details?

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Here are 5 related questions for further practice:

1. How would the prediction change if the common difference were slightly larger or smaller?
2. What is the population prediction for 2030 assuming the same arithmetic pattern?
3. Can you model this population growth using a geometric sequence instead?
4. What would be the effect of a decreasing common difference on the population growth?
5. How can we account for external factors like migration or birth rate changes in this model?

Tip: Arithmetic sequences are useful for modeling consistent growth patterns, but remember that real-world factors can cause fluctuations, making precise predictions difficult.