Number Series

श्रृंखला में प्रत्येक संख्या अगली संख्या के 5 गुने से घटाकर बनती है: 23×5-3=112, 112×5-23=537 (यह 637 होना चाहिए), 537×5-30=2655 (यह 3155 होना चाहिए), 2655×5-71=12604। 637 गलत है क्योंकि यह पैटर्न को तोड़ता है।

Pattern: each term = previous term × 2 + 1. 15 × 2 + 1 = 31, 31 × 2 + 1 = 63, 63 × 2 + 1 = 127, 127 × 2 + 1 = 255, 255 × 2 + 1 = 511. The missing number is 63.

Pattern: each term = previous term ÷ 2. 928 ÷ 2 = 464, 464 ÷ 2 = 232, 232 ÷ 2 = 116, 116 ÷ 2 = 58, 58 ÷ 2 = 29. The missing number is 29.

Pattern: multiply by position, then add 1. 3.5 × 1 + 1 = 4.5, but that doesn't match. Let's try: 3.5 - 1 = 2.5, 2.5 × 1 + 0.5 = 3, 3 × 2 + 0 = 6, 6 × 3 + 2 = 20, 20 × 4 + 15 = 95. Another view: (n² + 1) pattern: 3.5×1+1=4.5 no. Try: term(n) = term(n-1) × n - (n-2). 2.5 = 3.5×1-1, 3 = 2.5×2-2, 6 = 3×3-3, 20 = 6×4-4, 95 = 20×5-5. Missing: 6.

Pattern: each term = previous term × 1.5. 4.5 × 1.5 = 6.75, 6.75 × 1.5 = 10.125 (not 13.5). Let's check forward: 6.75 × 2 = 13.5, 13.5 × 2.5 = 33.75, 33.75 × 3 = 101.25. The multipliers increase by 0.5: 1.5, 2, 2.5, 3. Working backward: 4.5 × 1 = 4.5. The missing number is 4.5.

The pattern in the differences increases by 4.5 each time: 2, 6.5, 11, 15.5, 20. Adding these to each term: 12+2=14, 14+6.5=20.5, 20.5+11=31.5, 31.5+15.5=47, 47+20=67. The missing term is 31.5.

The series follows the pattern: multiply by increasing integers starting from 1, then add increasing integers starting from 1. Specifically: 4×1+1=5, 5×2+5=15, 15×3+4=49, 49×4+5=201, 201×5+6=1011, 1011×6+7=6073. The pattern for additions from 5 onwards is +5, +4, +5, +6, +7. If the second term were correct, we'd have: 5×2+5=15. But the starting value is 5 when it should follow 4×1+1=5 as the base. The issue is that 5 appears twice - as both the second term and part of the calculation. The term 5 should be the starting point for the pattern to work consistently.

This is an alternating series: Series 1 (odd positions): 126, 128, 130 (+2 pattern). Series 2 (even positions): 139, 138, 135 (should be 137 to maintain -1 pattern: 139-1=138, 138-1=137). The number 138 breaks the alternating pattern - the even position series should decrease by 1 each time, so after 138 should come 137, not 135. Thus 138 is the wrong number in its position.

The series has two interleaved patterns. The odd positions (77, 79, 81, ?) increase by 2 each time. The even positions (14, 16, 09, ?) have the pattern: 14 → 16 (+2), 16 → 09 (-7), so the next should be 09 + 7 = 16. Wait, let me reconsider. Actually: 14-16-09 form a pattern where digits sum: 1+4=5, 1+6=7, 0+9=9. The next in sequence 83-11 continues: odd position is 81+2=83, and 11 follows the digit sum pattern. The complete sequence is 77, 14, 79, 16, 81, 09, 83, 11.

Find the pattern by calculating differences between consecutive terms: 27→29 (+2), 29→38 (+9), 38→61 (+23), 61→105 (+44), 105→178 (+73), 178→284 (+106). The second differences are: 7, 14, 21, 29, 33. The pattern should be 7, 14, 21, 28, 35. The 73 after 105 breaks the pattern, so 178 is wrong.

The pattern is: multiply previous term by (n+1)^2 where n is the position. 1×4=4, 4×9=36, 36×16=576, 576×25=14400. The multipliers are squares of consecutive natural numbers starting from 2: 2^2=4, 3^2=9, 4^2=16, 5^2=25.

The series should be: 37 (+4) 41 (+2) 43 (+6) 49. The pattern of adding 4, 2, 6 suggests next should be +4, giving 53. The second 49 (option C) is the wrong number that breaks the prime number pattern and the addition pattern.

The pattern involves dividing by successively smaller numbers: 1799 ÷ 6 ≈ 299.83 (rounds to 299), 299 ÷ 5 = 59.8 (rounds to 60, but given as 59), 59 ÷ 4 = 14.75 (rounds to 15, but given as 14), 14 ÷ 3.5 = 4, 4 ÷ 3 ≈ 1.33 (given as 1.5). The exact mathematical relationship is approximate.

The pattern is n² + 1: 1²+1=2, 2²+1=5, 3²+1=10, 4²+1=17, 5²+1=26, 6²+1=37. The number 28 breaks this pattern (it should be 26). All other numbers follow n² + 1 correctly.

The pattern is +22, +22, +21, +23, +22. The differences should all be 22: 121+22=143, 143+22=165, 165+22=187 (not 186), 187+22=209, 209+22=231. The number 186 is one less than expected.