Find the next term of the sequence 10, 74, 202, 394, ___

Concept: In a number series where the differences between consecutive terms themselves increase by a fixed amount at each step (i.e. the second difference is constant), the terms follow a quadratic pattern — the sequence can be written as a closed-form function of an increasing base number once that form is identified from the given terms.

Application:

1. Observing the given terms, they fit the form 2 × (n² + 1): 10 = 2 × (2² + 1), 74 = 2 × (6² + 1), 202 = 2 × (10² + 1), 394 = 2 × (14² + 1).

2. The base numbers 2, 6, 10, 14 increase by 4 at each step, so the next base number is 14 + 4 = 18.

3. Substitute n = 18 into the rule: 2 × (18² + 1) = 2 × (324 + 1) = 2 × 325 = 650.

Cross-check: The plain differences between terms are 74 − 10 = 64, 202 − 74 = 128, and 394 − 202 = 192 — these themselves increase by a constant 64 each time (128 − 64 = 64, 192 − 128 = 64), confirming the quadratic pattern independently. Continuing that constant growth gives the next difference as 192 + 64 = 256, so the next term is 394 + 256 = 650 — the same value obtained above.

Result: the next term of the sequence is 650.