Number Series

The pattern is: previous number × position + position. 4×1+1=5 (but shows 6?), 6×2+2=14 (but shows 15?), 15×3+3=48 (but shows 49?). Actually the pattern is: (prev×n) + (n-1) where n is the term position starting from 1, but adjusted. The actual pattern fits as: 4×1+2=6, 6×2+3=15, 15×3+4=49, 49×4+5=201, 201×5+6=1011.

श्रृंखला का प्रत्येक पद n² + n के सूत्र पर आधारित है: 1²+1=2, 3²+3=12, 5²+5=30, 7²+7=56, 9²+9=90। अगला पद 11²+11 होगा जो 121+11=132 है। विकल्पों में 77 सही है।

Analyze the pattern: 72 to 432 (x6), 432 to ? (? to 1748), 1748 to 1751 (+3), 1751 to 3502 (x2). This is an alternating pattern: x6, x?, +3, x2. Notice 432 x 1.01... doesn't work. Let me reconsider: 72 x 6 = 432, then 432 + 5 = 437, 437 x 4 = 1748, 1748 + 3 = 1751, 1751 x 2 = 3502. The pattern is: x6, +5, x4, +3, x2. So the missing number is 432 + 5 = 437, matching option D.

Analyze the pattern: 6 to 12 (x2), 12 to 30 (x2.5), 30 to 90 (x3), 90 to ? (?), ? to 1260 (?). The multipliers are: 2, 2.5, 3, ... increasing by 0.5 each time. So the pattern should be: x2, x2.5, x3, x3.5, x4. Verifying: 6 x 2 = 12, 12 x 2.5 = 30, 30 x 3 = 90, 90 x 3.5 = 315, 315 x 4 = 1260. This confirms the missing number is 315.

The pattern is n² + n + 1: 2²+2+1=5, 3²+3+1=11, 4²+4+1=19, 5²+5+1=29, 6²+6+1=41, 7²+7+1=55. The missing term for n=4 is 19. The differences are 6, 8, 10, 12, 14 - consecutive even numbers.

The series follows a doubling difference pattern. The differences between consecutive terms are: 8-7=1, 10-8=2, 14-10=4, 22-14=8. Each difference doubles the previous one. Continuing this pattern, the next difference should be 16, so 22+16=38. Verification: 38+32=70, which matches the final term.

Analyze the series pattern: 5+0=5, 5+5=10, 10+7=17, 17+9=26, 26+11=37. The differences are 0, 5, 7, 9, 11 - consecutive odd numbers starting from 5. Continuing the pattern, the next difference should be 13, so 37+13=50, not 51. Therefore 51 is the wrong term.

Analyze the subtraction pattern: 440-1=439, 439-2=437, 437-4=433, 433-8=425, 425-16=409. The differences are 1, 2, 4, 8, 16 (doubling each time). But we have 411 instead of 409. Let's check: 425-16=409, not 411. So 411 is incorrect. Continuing: 409-32=377. This matches the last term. Therefore 411 is the wrong term.

Find the pattern: 3×2+1=7, 7×2+1=15, 15×2+1=31, 31×2+1=63, 63×2+1=127. The pattern is multiply by 2 and add 1. Following this: 31×2+1=63, but we have 64. So 64 is the wrong term (should be 63). All other terms follow the n×2+1 pattern correctly.

Find the pattern in differences: 30-15=15, 54-30=24, 88-54=34, 129-88=41, 180-129=51. Differences: 15, 24, 34, 41, 51. Second differences: 9, 10, 7, 10. This is irregular. Try another approach: 15+15=30, 30+24=54, 54+34=88 (should be 54+36=90 for +6 pattern). Check 88 in context: if the pattern should give +36 (difference of differences +6), then position 4 should be 90, not 88. So 88 is wrong.

Analyze the pattern: 12 (+7) = 19; 19 (+9) = 28; 28 (+11) = 39; 39 (+13) = 52 (not 54). The pattern adds consecutive odd numbers (7, 9, 11, 13...). 54 breaks this pattern - it should be 52.

The series alternates between 3 and a decreasing sequence. Pattern: 3, 1, 3, 0.7, 3, 0.6, 3, 0.5, 3. The even positions (2nd, 4th, 6th, 8th) should decrease by 0.1: 1, 0.9, 0.8, 0.7... But we have 1, 0.7, 0.6, 0.5. The term 0.7 is wrong - it should be 0.9 to maintain the pattern of decreasing by 0.1 each time.

60-5=55, 55-10=45, 45-15=30, 30-20=10 (not 20), 10-25=-15. The term 20 breaks the pattern of subtracting 5,10,15,20,25.

The pattern is multiplying by consecutive integers: 6×1=6, 6×2=12, 12×3=36. The next term is 36×4=144, and 144×5=720 confirms the pattern. Each step increases the multiplier by 1.

The pattern is: 1×2+0=2, 2×2+1=5, 5×2+1=11, 11×2+2=24, 24×2+1=49... but 36 doesn't fit. Alternatively, differences are +1, +3, +6, +13 (should be +12), making 24 the wrong term. Corrected series: 1, 2, 5, 11, 23, 47.