Question: A soil scientist measures the nutrient content in five soil samples taken at increasing depths, which form an arithmetic sequence. If the average nutrient level across the samples is 22 units, and the difference between the highest and lowest nutrient levels is 16 units, what is the nutrient level at the third sample? - Coaching Toolbox

Understanding the Context

From the problem:

• The average nutrient level across the five samples is 22 units.
  
• The difference between the highest and lowest nutrient levels is 16 units.

Let’s break down both pieces of information.

Step 1: Confirm the average

Since the sequence is arithmetic and symmetric, the mean is equal to the middle term—the third sample’s nutrient level, which is a.
Thus:
a = 22

Step 2: Use the range to find the common difference

The lowest nutrient level is a – 2d, and the highest is a + 2d.
The difference between them is:
(a + 2d) – (a – 2d) = 4d = 16
So:
4d = 16 → d = 4

Key Insights

Step 3: Find the nutrient level at the third sample

We already know a = 22, and the third sample’s nutrient level is a, which is 22.

Why the third term matters

In any arithmetic sequence, the middle term (third in a five-term sample) is both the average and the central value. This makes it highly valuable for identifying baseline fertility or deficiency markers. Here, despite variation with depth, the third sample’s nutrient level remains a reliable indicator of typical conditions across the profile.

Conclusion:
By modeling soil nutrient data as an arithmetic sequence, a researcher confirmed that the third sample’s nutrient level—acting as the average—equals 22 units. This demonstrates how mathematical patterns enhance interpretation of environmental data, guiding smarter decisions in farming and soil conservation.

Final Thoughts

Keywords: soil nutrient analysis, arithmetic sequence soil samples, average soil nutrients, nutrient level third sample, soil scientist measurement, soil fertility patterns