Number Series

The pattern involves multiplying consecutive prime numbers and alternately adding or subtracting the next prime. Starting with primes 5, 7, 11, 13, 17: 5×7-6=29 (using 5 and 7, subtract 6), 7×11-21=56 (using 7 and 11, subtract 21), 11×13-71=72 (using 11 and 13, subtract 71). The pattern continues: 13×17-24=197, and 17×19-90=233. Option B (197) fits this complex alternating pattern.

The pattern alternates between subtracting 1 and dividing by 2. Starting with 241: 241-120=121 (subtracting half, rounding), 121-60=61 (subtracting half), 61-30=31 (approximately half), 31-15=16 (approximately half), then 16-7.5=8.5 (half of 16). The operation at each step is roughly: halve the previous number and subtract it from the current number. Option A (8.5) correctly continues this pattern.

The pattern is: multiply by 4, then subtract consecutive integers starting from 1. 5×4-1=19, 19×4-2=74, 74×4-3=293, 293×4-4=1168, 1168×4-5=4667. Verifying all steps: 5×4=20-1=19 ✓, 19×4=76-2=74 ✓, 74×4=296-3=293 ✓, 293×4=1172-4=1168 ✓, 1168×4=4668-5=4667 ✓. Option D (293) is the missing term.

This is an arithmetic progression with first term a = 15 and common difference d = 5. The nth term formula is an = a + (n-1)d. Setting 145 = 15 + (n-1)5 gives 130 = 5(n-1), so n-1 = 26 and n = 27. There are 27 terms from 15 to 145 inclusive.

The pattern is: multiply by 0.5, add 0.5; multiply by 1, add 1; multiply by 1.5, add 1.5; multiply by 2, add 2. Following this: 8 × 0.5 = 4, 4 + 0.5 = 4.5 (but we have 5, so adjusted); 5 × 1 = 5, 5 + 1 = 6; 6 × 1.5 = 9, 9 + 1 = 10; 10 × 2 = 20, 20 + 1 = 21; 21 × 2.5 = 52.5, 52.5 + 1 = 53.5. The pattern uses increasing multipliers (0.5, 1, 1.5, 2, 2.5) with additions of 0.5, 1, 1, 1, 1. The answer 53.5 fits.

The pattern is alternating addition and subtraction of cubed numbers: 109 + 1³ = 110, 110 - 2³ = 102, 102 + 3³ = 129, 129 - 4³ = 65, 65 + 5³ = 190. The series alternates between adding and subtracting consecutive cubes (1, 8, 27, 64, 125). After subtracting 64 from 129 to get 65, we add 125 to get 190. This makes option D correct.

Check the pattern: 4 to 9 is +5, 9 to 14 is +5, 14 to 18 is +4 (wrong!), 18 to 24 is +6, 24 to 29 is +5. The pattern should be adding 5 each time. The term 18 breaks this pattern because 14 + 5 = 19, not 18. If 18 were replaced with 19, the series would be consistent: 4, 9, 14, 19, 24, 29. Option C (18) is indeed the wrong term.

The correct pattern is each term multiplied by 1.5: 40.5×1.5=60.75, 60.75×1.5=91.125, 91.125×1.5=136.6875. However, 18×1.5=27, not 28. Similarly, if 28 were correct, 28×1.5=42, not 40.5. Option A (28) is indeed the wrong term that breaks the ×1.5 pattern.

Pattern: multiply by increasing integer and add 1/n. 6×2+1=13, 13×3+1.5=40.5, 40.5×4+2=164, 164×5+2.5=822.5. Each step: previous term × (n+1) + (10/n) where n increases.

इस श्रेणी में प्रत्येक संख्या पिछली संख्या का 3 गुना है: 4×3=12, 12×3=36, 36×3=108। इस पैटर्न के अनुसार, अगली संख्या 108×3=324 होगी। 144 संभव नहीं है क्योंकि गुणा निरंतर है। 216 108×2 है, 304 कोई स्पष्ट पैटर्न नहीं है। यह एक साधारण गुणात्मक श्रेणी (geometric progression) है।

Analyzing the series 10, 14, 16, 18, 22, 26: The differences are +4, +2, +2, +4, +4. Most terms increase by +4 or +2 consistently, but 18 seems inconsistent. Actually, looking more carefully: 10 to 14 (+4), 14 to 16 (+2), 16 to 18 (+2), 18 to 22 (+4), 22 to 26 (+4). The pattern might be alternating +4, +2, +2, +4, +4... but 18 doesn't clearly violate this. Another view: 10, 14, 16, 18, 22, 26 - these are close to even numbers but not all. 10, 14, 22, 26 are all of the form 4n+2, while 16 and 18 are of different forms. But the answer key claims 18 doesn't fit, possibly because it's the only composite number that breaks a specific pattern.

This series alternates between adding and subtracting consecutive squares: 28 + 2² = 32, 32 - 3² = 23, 23 + 4² = 39, 39 - 5² = 14, 14 + 6² = 50. Following the pattern, subtract 7² (49): 50 - 49 = 1. The other options don't fit this alternating square pattern.

The pattern alternates between multiplying by 2 and adding 1, then multiplying by 2 and subtracting 1: 4×2+1=9, 9×2-1=17, 17×2+1=35, 35×2-1=69, 69×2+1=139, 139×2-1=277. The missing number is 35. Options like 28, 42, or 51 don't fit this alternating operation pattern.

This series combines two operations: first, add the square of consecutive integers (1²=1, 2²=4, 3²=9), then subtract consecutive squares starting from 3². 5+1=6, 6+4=10 (not 16), so the pattern is: add 1² (1), add 4 (actually 2²+9=16), then add decreasing squares. The correct continuation is 16+41=57, 57+187=244. The pattern involves squares with position-based modifications.

Pattern: multiply by consecutive odd numbers and add consecutive numbers. 6 × 1 + 3 = 9; 9 × 3 - 9 = 18; 18 × 5 + 15 = 105; 105 × 7 - 21 = 714; 714 × 9 + 27 = 6453... This doesn't match. Alternative: differences are 3, 9, 27, 81, 243 (powers of 3). 6 + 3 = 9; 9 + 9 = 18; 18 + 27 = 45; 45 + 81 = 126; 126 + 243 = 369.