Solved Examples

Example 1: Find the next term: 5, 11, 19, 29, 41, ? - First differences: 6, 8, 10, 12 → these increase by 2 each time (arithmetic pattern of differences) - Next difference = 14 → next term = 41 + 14 = 55

Example 2: Find the missing term: 2, 9, 28, 65, 126, ? - Compare with cubes: 1³+1=2, 2³+1=9, 3³+1=28, 4³+1=65, 5³+1=126 → pattern is n³+1 - Next term (n=6): 6³+1 = 216+1 = 217

Example 3: Find the odd one out: 3, 5, 11, 14, 17, 21 - All except 14 are odd numbers (also, ignoring that, 3,5,11,17 are primes while 14=2×7 and 21=3×7 are composite — but the cleanest single pattern is odd vs. even). 14 is the only even number in the list. Answer: 14 is the odd one out

Example 4: Find the next term: 4, 6, 12, 14, 28, 30, ? - Pattern alternates ×2 and +2: 4×1.5=6? Let’s check operations: 4→6 (+2), 6→12 (×2), 12→14 (+2), 14→28 (×2), 28→30 (+2), so next operation is ×2: 30×2 = 60

Example 5 (Find the wrong number — n²−1 pattern with a planted error): Find the wrong number in the series: 3, 8, 15, 24, 34, 48. - Compare against n² − 1 for n = 2, 3, 4, 5, 6, 7: 3, 8, 15, 24, 35, 48. - The given series has 34 in the 5th position instead of 35. Answer: 34 is the wrong number (it should be 35)

Example 6 (Second-order arithmetic series): Find the next term: 6, 12, 21, 33, 48, ? - First differences: 6, 9, 12, 15 — these themselves increase by a constant 3 each time (second-order arithmetic pattern). - Next difference = 15 + 3 = 18 → next term = 48 + 18 = 66 Answer: 66