Number Series

श्रृंखला दो अंतर्निहित क्रमों से मिलकर बनी है। विषम-स्थान (1,3,5,7): 2, 4, 7, 11 — अंतर: 2, 3, 4 — अगला अंतर 5 → 9वां पद = 11+5 = 16। सम-स्थान (2,4,6,8): 15, 47, 118, ? — पैटर्न: 15=2×7+1, 47=4×11+3, 118=7×16+6, ?=11×22+10 = 252। यहां 7,11,16,22 में अंतर: 4,5,6 हैं और 1,3,6,10 त्रिकोणीय संख्याएं हैं। उत्तर: 252, 16।

The pattern is: subtract consecutive terms, then multiply the difference by 5 to get the next difference. The differences are 12-22=-10, 16-12=4, 41-16=25, 1465-41=1424. Notice that each difference is 5 times the previous difference's square root: 4 = 2 × 2 (where 2²=4), 25 = 5 × 5 (where 5²=25), 1424 = ? × ?. Continuing the pattern: the next difference should relate to 25, and 25² = 625, with 5 × 25 = 125. The pattern works as: each term comes from the next term through the rule: term = next_term - (diff). For 16 to 41: diff = 25. For 41 to 1465: diff = 1424. The pattern: differences grow rapidly. Working backwards: 1465 - ? = 41 gives ? = 1424. 41 - ? = 16 gives ? = 25. 16 - ? = 12 gives ? = 4. 12 - ? = 22 gives ? = -10. The missing term before 1465 satisfies: ?² = 32400, so ? = 180.

The pattern follows: x2, x2+0, x2+7.5, x2+7.5, x2+75, x2+75. Starting with 3: 3x2=7.5, 7.5x2=15, 15x2+7.5=37.5, 37.5x2=75, 75x2+75=225, 225x2=450. The term 175 breaks the pattern - it should be 225. The series alternates between pure doubling (x2) and doubling with adding 7.5 (x2+7.5).

Analyze the pattern: 20 = 17 × 1 + 3, 46 = 20 × 2 + 6, 150 = 46 × 3 + 12. The pattern is: next term = current term × n + (3 × 2^(n-1)), where n increases by 1 each time. Continuing: 150 × 4 + 24 = 624, not 600. The term 600 breaks the pattern and should be 624. The correct term after 600 would be 624 × 5 + 48 = 3168, not 3015. Checking: 3015 = 624 × 5 - 105, doesn't fit. Let's re-verify: 17 × 1 + 3 = 20 ✓, 20 × 2 + 6 = 46 ✓, 46 × 3 + 12 = 150 ✓, 150 × 4 + 24 = 624 (given is 600) ✗. So 600 is wrong.

Looking at the series 240, 229, 222, 213, 208, 205, 204, let's find the pattern: 240-229=11, 229-222=7, 222-213=9, 213-208=5, 208-205=3, 205-204=1. The differences are 11, 7, 9, 5, 3, 1. This doesn't follow a clear pattern. Let me check alternating terms: 240→222→208→204 gives -18, -14, -4 (not consistent). Or 229→213→205 gives -16, -8 (halving). The pattern might be based on digit manipulation or position. Actually, looking at it differently: subtract consecutive digits? No. Let's try: 240, 229 (240-11), 222 (229-7), 213 (222-9), 208 (213-5), 205 (208-3), 204 (205-1). The subtractions are 11,7,9,5,3,1. These are odd numbers in descending order but 7,9,5 is not descending (should be 11,9,7,5,3,1). The term 222 breaking this pattern (after 229 should be 220 if pattern held). But actually 229-222=7 and the sequence of subtractions would be 11,9,7,5,3,1 if perfect. We have 11,7,9,5,3,1. So 7 and 9 are swapped - making 222 the odd one (should have come after 9 subtraction from 229, giving 220). Therefore 222 doesn't fit the pattern.

The series follows the pattern: add previous difference + increasing offset. Differences are -6, -3, +2, +11, +28. The pattern in differences is +3, +5, +9, +17 (gaps doubling: 2, 4, 8). Expected next difference: +17+16=33, giving 76+33=109. But the series has 135, which breaks this pattern. Alternative: (n-1)+(n-2)+k where k increases. 37+35=72 needs +10 to get 82, not 76. 135 is clearly wrong.

The pattern alternates between ÷2 and ×3: 20 ÷ 2 = 10, 10 × 3 = 30, 30 ÷ 4 = 7.5, 7.5 × 5 = 37.5. Following the pattern: 37.5 ÷ 6 = 6.25. The divisors/multipliers increase by 1 each step (2, 3, 4, 5, 6).

The differences are: 209 - 200 = 9 = 3², 237 - 209 = 28 = 3³ + 1, 302 - 237 = 65 = 4³ + 1. The next difference should be 5³ + 1 = 126. So 302 + 126 = 428.

Working backwards from 53: differences are 55 (108-53), 77 (185-108), 99 (284-185), 121 (405-284). These are consecutive odd numbers. Before 53, subtract 7: 53 - 7 = 46. But 46 isn't an option. Testing forward from each option: 20 + 33 = 53, 53 + 55 = 108, 108 + 77 = 185, 185 + 99 = 284, 284 + 121 = 405. The pattern is odd numbers starting from 33: 33, 55, 77, 99, 121 (increasing by 22 each time).

The sequence 102, 101, 98, 93, 86, 74, 66, 53 follows a pattern where the difference between consecutive terms decreases by 2 each time: 102-101=1, 101-98=3, 98-93=5, 93-86=7, 86-74=12 (should be 9), 74-66=8 (should be 11), 66-53=13. The pattern should be differences of 1, 3, 5, 7, 9, 11, 13. This would make 86-9=77, not 74. Therefore 74 is the wrong number.

The series follows a pattern where the first number increases by 2 (5, 7, 9, ?, 13), so the missing term should start with 11. The second number is always 12. The third number also increases by 2 (6, 8, 10, ?, 14), so it should be 12. The fourth number is always 1. Therefore, the missing term is 11-12-12-1.

Sequence: 0, 3, 14, 39. First differences: 3-0=3, 14-3=11, 39-14=25. Second differences: 11-3=8, 25-11=14. Third difference: 14-8=6 (constant). This is a cubic sequence: n³-n. For n=4: 4³-4=60, not in options. Alternative: differences follow 3, 11, 25 which are 2²-1, 3²+2, 5². Next could be 7²+6=55 (if pattern n²+(n-1)) or 45 (if odd numbers). 39+45=84 works.

The series shows a pattern of two repetitions followed by subtracting 13: 53, 53 (repeat), 53-13=40, 40 (repeat), 40-13=27, 27 (repeat). Following this pattern, the next term is 27-13=14. The sequence continues: 14, 14, 1, 1, etc.

This series has a constant third difference of 19. The first differences are 20, 39, 77, 134, and the next is 210 (134+76 where 76 is 57+19). So 310+210=520.

Working backwards, the series has a constant third difference of 6. With ?=-1, the differences are: -1→5 (+6), 5→23 (+18), 23→59 (+36), 59→119 (+60), 119→209 (+90). Second differences: 12, 18, 24, 30. Third differences: 6, 6, 6 (constant).