Number Series

The pattern is n² + n: 2²+2=6, 4²+4=20, 7²+7=56, 10²+10=110, 13²+13=182. The numbers 5, 22, 108, 175 don't fit this pattern. Specifically, 108 should be 110 (10²+10), making it the wrong number.

The series alternates between a number and its cube: 2^3=8 (not 7), 3^3=27, 4^3=64, 5^3=125. The first pair is wrong - it should be 2 and 8, not 2 and 7. Therefore 7 is the wrong number in the series.

The pattern is: -1, -4, -2, -3, -3, -2. Starting from 20: 20-1=19, 20-4=16, 19-2=17, 16-3=13, 17-3=14, 13-2=11. This creates 20, 20, 19, 16, 17, 13, 14, 11.

The pattern involves differences: 6-5=1, 15-6=9, ?-15=25, 89-?=49, 170-89=81, 291-170=121. The differences are 1, 9, 25, 49, 81, 121 (squares of odd numbers). So 49+40=89.

This follows a pattern of: multiply by increasing consecutive integers starting from 2, then add consecutive integers starting from 2. 8×2+2=18, 18×3+3=57, 57×4+4=232. Following this pattern: 232×5+5=1165. The multiplier increases by 1 each time (2, 3, 4, 5), and the added number also increases by 1 each time (2, 3, 4, 5). This creates an accelerating series where each term depends on the previous term through multiplication and addition.

The pattern multiplies by consecutive prime numbers: 4x2=8, 8x3=24, 24x5=120, 120x7=840, 840x11=9240. The term 168 breaks this pattern - it should be 120 (24x5, not 24x7). 168 is the odd number.

Pattern: 4×1+5=9, 9×2+8=26, 26×3+6=84, 84×4+6=342, 342×5+6=1716... But checking carefully: 4×1=4, 4+5=9; 9×2=18, 18+8=26; 26×3=78, 78+6=84; 84×4=336, 336+6=342; 342×5=1710, 1710+6=1716; 1716×6=10296, 10296+6=10302. Wait, 9 is wrong! The first term should be 4, not 9. Actually, rechecking: 4 to 9 (+5), 9 to 26 (×2+8), 26 to 84 (×3+6)... pattern breaks at start. 9 doesn't fit a clear pattern with 4.

Pattern: 150÷2=75, 75÷1.5=50, 50÷1.25=40, 40÷1.33≈30, 30÷1.2=25. Actually, check: 150→75 (÷2), 75→50 (÷1.5 or ×2/3), 50→40 (÷1.25 or ×4/5), 40→30 (÷1.33 or ×3/4), 30→25 (÷1.2 or ×5/6). Pattern is dividing by (n+1)/n: ÷2/1, ÷3/2, ÷5/4, ÷4/3 (should be ÷4/3=÷1.33 giving 30, correct from 40), then ÷6/5=÷1.2. The series is consistent - 40 doesn't fit? Wait, checking 50÷1.25=40, then 40÷1.33=30... the divisors are 2, 1.5, 1.25, 1.2... which are 2/1, 3/2, 4/3, 5/4, 6/5. But 50→40 is ×0.8=4/5, should be 4/3=1.33. So 40 should be 50÷(4/3)=37.5, not 40. Actually, let me re-verify: 150÷2=75, 75÷1.5=50, 50÷1.333...=37.5 not 40, 37.5÷1.25=30, 30÷1.2=25. So 40 is wrong.

The differences between consecutive terms are 122, 169, 225, 289, 361, 441. Except for the first difference (122), all others are perfect squares of odd numbers: 13²=169, 15²=225, 17²=289, 19²=361, 21²=441. The first difference should be 11²=121, meaning the first term should be 123-121=2, not 1.

The differences between consecutive terms are: 8793-8450=343=7³, 9009-8793=216=6³, 9135-9009=126, 9198-9135=63. The pattern of differences should be 7³, 6³, 5³=125, 4³=64. The difference 126 breaks the pattern (should be 125), making 9135 the wrong number. If 9009+125=9134, then 9134+64=9198, which fits.

The series should follow: ×1+1, ×2-0.5, ×2+0.5, ×2-1, ... Starting from 3: 3×1+1=4, 4×2-0.5=7.5, 7.5×2+0.5=15.5 (not 19), 15.5×2-1=30, 30×2+... The value 19 breaks the pattern.

Finding the pattern in differences: 39-15=24, 74-39=35, 122-74=48. The second differences are: 35-24=11, 48-35=13. The second differences increase by 2 each time (11, 13, 15, 17...). So next first difference should be 48+15=63. Adding to last term: 122+63=185. Verification: Next difference would be 63+17=80, giving 185+80=265, which matches the final term. Option A (171) would require difference of 49, breaking the pattern. Option C (151) would require difference of 29, breaking the pattern. Option D (165) would require difference of 43, breaking the pattern. Option B (185) correctly fits.

The pattern is n²-1 starting from n=0: 0²-1=-1, 1²-1=0, 2²-1=3, 3²-1=8, 4²-1=15. The next term is 5²-1=24. Alternatively, the differences increase by 2 each time: +1, +3, +5, +7, so next is +9 giving 24.

Analyze the multiplication pattern: each term should be double the previous term (×2). 15×2=30, but the series shows 28. 30×2=60 (correct), 60×2=120 (correct), 120×2=240 (correct). The number 28 breaks this doubling pattern - it should be 30. Therefore, 28 is the wrong number in the series.

The pattern alternates between multiplication and addition: 9×3=27, then 27+4=31, then 31×5=155, then 155+6=161, then 161×7=1127. Following this pattern, the next operation is addition by 8: 1127+8=1135. The multipliers and addends increase by 1 each step (×3, +4, ×5, +6, ×7, +8).