I meet with my friend on each Sunday. On first Sunday I met with him at 12.30…

When the successive differences of a sequence themselves form an arithmetic progression (that is, the second differences are constant), the sequence follows this constant-second-difference pattern. The next term is found by extending the pattern of differences, not by adding a fixed interval.

1. Convert every recorded meeting time to minutes measured from 12:00 noon: Sunday 1 = 12:30 PM = 30 minutes; Sunday 2 = 1:20 PM = 80 minutes; Sunday 3 = 2:30 PM = 150 minutes; Sunday 4 = 4:00 PM = 240 minutes.

2. Find the first differences between consecutive Sundays: 80 minus 30 = 50 minutes; 150 minus 80 = 70 minutes; 240 minus 150 = 90 minutes.

3. Find the second differences, that is, the differences between the first differences: 70 minus 50 = 20 minutes; 90 minus 70 = 20 minutes — a constant value.

4. Since the second difference is constant at 20 minutes, extend the pattern: the next first difference is 90 plus 20 = 110 minutes.

5. Add this to Sunday 4's value: 240 plus 110 = 350 minutes measured from 12:00 noon.

6. Convert 350 minutes back into a clock time: 350 minutes equals 5 hours and 50 minutes, so the fifth meeting is at 5:50 PM.

As an independent check, fit the values with the standard formula for a sequence with constant second difference d and first term-to-term gap a: T(n) = T(1) + (n - 1) x a + [(n - 1)(n - 2) / 2] x d, using a = 50 and d = 20. For the fifth Sunday (n = 5): T(5) = 30 + 4 x 50 + 6 x 20 = 30 + 200 + 120 = 350 minutes, the same value obtained by extending the difference pattern, confirming 5:50 PM.