Number Series

The pattern follows: ×7-0.2, ×5+0.3, ×5.5-1, ×6+0.8, ×7-1 (approximately). Checking: 2.3×7 = 16.1, but we have 15.9 (difference -0.2). 15.9×5 = 79.5+0.3 = 79.8. 79.8×5.5 = 438.9-15.9 = 423. 423×6 = 2538+36 = 2574. 2574×7 = 18018-49 = 17969, but we have 18067. The term 79.8 fits the pattern while others have inconsistencies, or 79.8 should have been calculated differently.

The pattern alternates between two operations: ×2 - 4 and ÷2 - 1. Starting from 73: 73 × 2 - 4 = 146 - 4 = 142 ✓, 142 ÷ 2 - 1 = 71 - 1 = 70 ✗ (should be 45). Let me reconsider. Alternative: 73 - ? = 142? No, that's increase. Maybe two interleaved sequences: positions 1,3,5,7: 73, 45, 28, 16. Differences: 73-45=28, 45-28=17, 28-16=12. Second differences: 11, 5. Not clear. Positions 2,4,6: 142, 88, 52. Differences: 142-88=54, 88-52=36. Ratio 54:36 = 3:2. If the pattern for odd positions follows similar ratio: 28:? = ? First sequence should have 45 at position 3. Let me check if 45 is wrong. If 45 were replaced with 47: 73-47=26, 47-28=19. Not helpful. Actually, the pattern might be: each number is derived from the previous. 73×2-4=142, 142÷3-2=45.3? No. Let me try: 73, 142, 47, 94, 31, 62, 21. This would be ×2, ÷3-1, ×2, ÷3-1, ×2, ÷3-1. If 45 is replaced with 47, the pattern is closer. The claimed answer 45 being 'wrong' makes sense if it should be 47 to maintain the alternating pattern.

The pattern is 'add consecutive odd numbers starting from 1'. 5+1=6, 6+3=9, 9+5=14, 14+7=21, 21+9=30. The increments are 1, 3, 5, 7, 9 (consecutive odd numbers). Option B (30) fits this pattern. 28, 29, 31 don't continue the sequence correctly.

The series alternates between two patterns: odd positions increase by 2 (5→7→9→11) and even positions decrease by 4 (18→14→10→6). Position 5 should be 9, but it's 10, making 10 the wrong number. 14 and 18 correctly follow the even-position decreasing pattern.

The pattern is: 4, then 4²+1=17, then 17²+1=290, then 290²+1=84101. Check: 17² = 289, 289+1 = 290. 290² = 84100, 84100+1 = 84101. This matches option B.

Analyze the pattern: 3→7 (+4), 7→16 (+9), 16→32 (+16), 32→56 (+24), 56→93 (+37), 93→142 (+49). The differences themselves have a pattern: 4, 9, 16, 24, 37, 49. The perfect squares 4, 9, 16 suggest the pattern should be 4, 9, 16, 25, 36, 49 (consecutive squares starting from 2²). If we replace 56 with 57 (making 32→57 +25), the pattern becomes: +4, +9, +16, +25, +36, +49. So 56 is the wrong number.

Check differences: 6→15 (+9), 15→35 (+20), 35→63 (+28), 63→143 (+80), 143→221 (+78), 221→323 (+102). This is irregular. Try alternate pattern or factor-based: 6=2²+2, 15=4²-1, 35=6²-1, 63=8²-1. Pattern: even squares plus/minus constant. Following this: 10²-1=99 (not 143), so 143 breaks it. Or: 6=2×3, 15=3×5, 35=5×7, 63=7×9, next should be 9×11=99 (not 143). So 63 is wrong. Answer 63 is correct.

Analyze the pattern: 5→55 (×11), 55→495 (×9), 495→3465 (×7), 3465→17325 (×5). The multipliers decrease by 2 each time: 11, 9, 7, 5. Following this pattern, the next multiplier should be 3: 17325×3=51975, not 34650. So 34650 is wrong. The correct series should continue with ×3, then ×1.

Option C is correct. The series alternates between two sequences: 3, 8, 10, 12 (increasing by 2-5 pattern) and the '1' entries. The second '1' at position 5 breaks what should be an odd-even alternating pattern where odd positions should have one sequence and even positions another. Position 5 (odd) having '1' when it should continue the first sequence makes it the wrong number.

The series is: 10, 22, 46, 94, 190, ?. Pattern: ×2+2. 10×2+2=22, 22×2+2=46, 46×2+2=94, 94×2+2=190, 190×2+2=382. The missing term is 382.

The series follows: +4, +10, +10, +16 (32 to 36 to 46 to 56 to 72). The correct pattern should be: +4, +8, +12, +16, making it 32, 36, 44, 56, 72. Therefore 46 breaks the arithmetic progression of differences. Each difference increases by 4.

The series is: 5×2-4=6, 6×2+2=14, 14×3-1=41, 41×2.5+?=? Actually: 5, 6, 14, 41, 105, 231. Pattern: differences are 1, 8, 27, 64 which are 1³, 2³, 3³, 4³. So next difference should be 5³=125, giving 105+125=230. But we have 231. If 231 is wrong term, then actual next should be 230. So 231 doesn't fit the pattern.

The pattern is: +1, +9, +25, +49, +81, +121... which are squares of odd numbers (1², 3², 5², 7², 9², 11²). Checking: 6+1=7, 7+9=16, 16+25=41, 41+49=90, 90+81=171 (not 154), 171+121=292. So 154 should be 171. The difference between consecutive terms follows odd squares.

The pattern is subtract increasing squares: 63 - 1² = 62, 62 - 1² = 61, 61 - 2² = 57, 57 - 4² = 49, 49 - 6² = 13. The squares being subtracted are 1, 1, 4, 16, 36 which follow the pattern 1², 1², 2², 4², 6². Alternatively, looking at differences: -1, -1, -4, -8, -27. The pattern involves subtracting consecutive squares from a sequence. The missing number is 61 - 4 = 57.

The series alternates between two interleaved sequences. Taking alternate terms: 9, (?), 11, 12 and 7, 9, 11, 13. The second sequence increases by 2: 7, 9, 11, 13. The first sequence increases by 1: 9, 10, 11, 12. Therefore the missing term is 10. This creates the pattern: 9, 7, 10, 9, 11, 11, 12, 13 where odd positions increase by 1 and even positions increase by 2.