Complete the series: 10, 100, 200, 310, ?

Concept: When the gaps between consecutive terms of a number series themselves increase by a fixed amount at every step, the series is a second-order (quadratic) arithmetic pattern: the first differences form their own arithmetic progression, so each new gap equals the previous gap plus that fixed increase.

1. Find the gap between each pair of consecutive terms: 100 − 10 = 90, 200 − 100 = 100, 310 − 200 = 110.

2. These gaps (90, 100, 110) themselves increase by a constant 10 at every step, confirming the second-order pattern.

3. So the next gap, from the 4th term to the 5th term, is 110 + 10 = 120.

4. Add this gap to the last given term: 310 + 120 = 430.

Cross-check: As an independent check, since the gaps have a constant second difference of 10, the nth term fits a quadratic term(n) = 5n² + 75n − 70 (because 2a = 10 gives a = 5). This gives term(1) = 10, term(2) = 100, term(3) = 200, and term(4) = 310 — matching every given term — and term(5) = 5(25) + 75(5) − 70 = 125 + 375 − 70 = 430, confirming the same answer.

So, the missing term is 430.