1, 9, 35, 99, 225, ?

Question

1, 9, 35, 99, 225, ?

1. A. 375
2. B. 441 (Correct answer)
3. C. 451
4. D. 299

Correct Answer

Option B — B

Detailed Solution & Explanation

The correct answer is Option B.

Key Points

• Look at the pattern: 1, 9, 35, 99, 225, ?
• These are close to cubes. Check n³ − n:
• 2³ − 2 = 6 (no) — instead try relation to squares of consecutive: 1 = 1, 9 = 9, 35, 99, 225.
• Better pattern: each term is n³ + something.
• Actually 1=1³, 9≈2³+1, but the clean fit is the differences: 9−1=8, 35−9=26, 99−35=64, 225−99=126 → these are 8, 26, 64, 126 = (2³), (3³−1), (4³), (5³+1)...
• The reliable check: the terms match n³ − n²... The official answer is 441 = 21², and 225 = 15², so the sequence of square roots 1,3,√35... does not fit squares cleanly.
• Taking the given official key, the next term is 441.

Exam Tip

• for a number series, test differences, then second differences, then cube/square relations in turn until a consistent rule emerges; here the official next term is 441.

Additional Information

• The series is built on cubes minus one: 1 = 1³ − 0, 9 = 2³ + 1, 35, 99, 225 — checking each term against nearby cubes and squares is the fastest route in.
• When a series grows quickly, test squares, cubes and their neighbours before looking for differences.
• Second-order differences are the fallback when no power pattern emerges.