A GPS tracker on a bird records its altitude over five consecutive hours as an arithmetic sequence. The sum of the altitudes is 1,000 meters, and the highest altitude is 240 meters. Find the common difference of the sequence. - Sterling Industries

A GPS tracker on a bird records its altitude over five consecutive hours as an arithmetic sequence. The sum of the altitudes is 1,000 meters, and the highest altitude is 240 meters. Find the common difference of the sequence.
Though not commonly discussed, this pattern appears increasingly relevant in tracking wildlife migration and bird behavior—especially as GPS technology becomes more precise and accessible. Over just five hours, birds may adjust altitude gradually, creating measurable patterns that mirror mathematical sequences. Recent interest in real-time wildlife monitoring reflects growing public curiosity about animal ecology, supported by widespread interest in environmental data tools.

The altitude data forms an arithmetic sequence—a sequence where each term increases or decreases by a consistent amount, called the common difference. Given the highest altitude is 240 meters, the sequence climbs to this peak before potentially declining. The sum of five terms in such a sequence equals 1,000 meters. Using the formula for the sum of an arithmetic sequence, we know:
Sum = (number of terms) × (first term + last term) / 2
But because the altitudes form a sequence peaking at 240, we understand the middle term—typically the third—represents the mean. With five terms, symmetry around the peak reveals that the third altitude equals the average, which is 1,000 ÷ 5 = 200 meters. Since the sequence peaks at 240, the altitudes rise steadily and then fall evenly.

To find the common difference (let’s call it d), let the first altitude be a. The sequence progresses as:
a, a + d, a + 2d, a + 3d, a + 4d
We know the third term: a + 2d = 200
And the fifth term: a + 4d = 240
Subtract the first from the second: (a + 4d) – (a + 2d) = 240 – 200 → 2d = 40 → d = 20
This consistent 20-meter rise confirms the sequence climbs steadily to the peak, then descends the same gradual rate afterward.