Number Series

Each term is multiplied by consecutive integers starting from 2: 2×2=4, 4×3=12, 12×4=48, 48×5=240, 240×6=1440. The multiplier increases by 1 each time (2, 3, 4, 5, 6). This is a multiplication-based progression, not a geometric progression since the ratio changes each step.

The pattern is: multiply by an odd number and add the same odd number. 85×1+1=86, 86×3+3=261, 261×5+5=1310, 1310×7+7=9177. Next: 9177×9+9=82602. Each step uses consecutive odd numbers (1,3,5,7,9).

The series alternates between squares and their square roots: 4 (2²), 196 (14²), 16 (4²), 169 (13²), ? (6²=36), 144 (12²), 64 (8²). The square roots follow the pattern: 2, 14, 4, 13, 6, 12, 8. The odd positions increase by 2 (2→4→6→8) and even positions decrease by 1 (14→13→12). The missing term is the square of 6.

Analyzing the pattern: 120 = 1×120, 440 = 4×110, 960 = 8×120, 1680 = 12×140. The second factors are: 120, 110, 120, 140. The differences are: -10, +10, +20. Following this pattern, next difference would be +30, giving 170. First factors: 1, 4, 8, 12 are increasing by 3, 4, 4, so next increase by 4 gives 16. So 5th term = 16 × 170 = 2720, which isn't an option. Let me try another approach: 120 + 320 = 440, 440 + 520 = 960, 960 + 720 = 1680. The additions are 320, 520, 720 (increasing by 200 each time). Next addition would be 920, so 1680 + 920 = 2600. This matches option A.

The series follows an alternating pattern with square numbers: subtract 1², add 2², subtract 3², add 4², subtract 5². Starting from 15: 15-1=14, 14+4=18, 18-9=9, 9+16=25, 25-25=0. The operations alternate between subtraction and addition while the squared numbers increase consecutively (1, 4, 9, 16, 25). Option A (36) incorrectly assumes only addition, while Option D (82) doesn't follow the pattern.

Check the pattern: 900 / 2.5 = 360, 360 / 3 = 120, 120 / 4 = 30, 30 / 5 = 6, 6 / 6 = 1. The pattern should be divide by increasing integers (2, 3, 4, 5, 6). But 900 / 2 = 450, not 360. If the series started with 1000: 1000/2.5=400 (close), or if first term was 900 and divisor was 2.5 (not consistent). The wrong term is 900 - it should be a number that when divided by 2.5 gives 360, which is 900, but the divisor should be 2 (not 2.5) for consistency. Option C is correct.

The pattern is alternating additions and subtractions following a Fibonacci-like sequence: +1, -8, +9, -14, -25. The magnitudes 1, 8, 9, 14, 25 don't follow a clear pattern, but 5 breaks the expected flow as it should be a positive term around 26 (5+25=30, then alternating back to positive would give ~26). The term -20 follows correctly from 5-25, but 5 itself is the anomaly as it disrupts the alternating sign pattern.

The series alternates between subtracting and adding: 169-155=14, 155+180=335 (doesn't fit), 180-136=44, 136+191=327, 191-125=66. Looking at odd positions: 169, 180, 191 (add 11, then 11). Even positions: 155, 136, 125 (subtract 19, then 11, doesn't fit). 155 breaks the pattern as even-position terms should be 155-19=136, 136-19=117, not 125.

The pattern is: ÷2, +0, ×3, +0, ÷2... but position 3 should be 9÷2=4.5 not 6. Checking the correct pattern from 12: 12÷2=6, 6+0=6, 6×3=18, 18÷4.5=4, 4×5=20... wait that doesn't work either. Let me try: 12-3=9, 9-3=6, 6×3=18, 18÷4=4.5, 4.5×5=22.5. This pattern works except position 4 should be 9÷2=4.5 not 6. So 9 at position 2 is wrong because it breaks the alternating pattern. Option D is correct.

Pattern: 20×2+5=45, 45×2+4=94, 94×2-8=180, 180×2-64=296, 296×2-127=465. The pattern is ×2+5, ×2+4, ×2-8, ×2-64, ×2-127... Position 4 should be 94×2+? Following the pattern +5, +4, the next should be +3: 94×2+3=191, not 180. So 180 is wrong. Option B is correct.

Starting from -1 doesn't work with the sequence. Try starting from 7: 7×1+1=8, 8×8-1=63, 63÷3-?=24... This doesn't yield a clear pattern. However, notice that if we remove -1, the pattern 7, 8, 63, 24, 215 has some relationship: 7+1=8, 8×8-1=63, 63÷3=21≈24, 24×9-1=215. The -1 doesn't fit any logical pattern starting the sequence, making it the wrong term. Option A is correct.

Pattern of differences: 8342-8173=169, 8173-7849=324, 7849-7320=529, 7320-6530=790. These are 13², 18², 23²... but 790 is not a perfect square. The next difference should be around 28²=784, giving 7320-784=6536, not 6530. So 6530 is wrong. Option B is correct.

Check each term with the pattern: n-2 × 2 + (n-2)/2 = current term. 6: base case. 9: 6×2 + 3 = 15 ✓. 15: 9×2 + 4.5 = 22.5 ✗ (should be 22.5). 22 should be 22.5. 51: 15×2 + 7.5 = 37.5 ✗ (should be 37.5). 99: If 22 were 22.5, then 22.5×2 + 11 = 56 ✗. The pattern breaks at 22, making it the wrong number.

श्रेणी 3, 5, 6, 11, 9, 17, 12, ? में दो interleaved sequences हैं। विषम स्थान (3, 6, 9, 12) में 3 का अंतर है। सम स्थान (5, 11, 17, ?) में 6 का अंतर है। इसलिए अंतिम संख्या = 17 + 6 = 23 होगी। यह alternating series है जहां दो अलग-अलग patterns एक-एक स्थान छोड़कर चलते हैं। 20 इस pattern में फिट नहीं बैठता।

Find the pattern: differences are 9, 11, 13, 15, 17, 18. The first five differences increase by 2 (consecutive odd numbers). The difference after 67 should be 17, giving 84. But the next difference shown is 18, not 17. Also, continuing the pattern: 84 + 17 = 101, not 102. The number 102 breaks the pattern.