Number Series

The pattern is: each term = n² + 1, where n starts from 1. 1² + 1 = 2, 2² + 1 = 5, 3² + 1 = 10, 4² + 1 = 17, 5² + 1 = 26, 6² + 1 = 37. The missing number is 5² + 1 = 26. Option A (20), C (25), and D (16) don't fit this pattern.

The series follows a factorial pattern: 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120. Each term is the factorial of its position. The next term is 6! = 720. This is a classic number series question testing recognition of factorial sequences.

The second differences from 22 onward are: 46-32=14, 73-46=27, 108-73=35, 158-108=50. The pattern 14, 27, 35, 50 doesn't follow a clear sequence. Testing if 19 should be 21 (giving +11, +15, +20, +28, +38, +50 where differences increase by 4, 5, 8, 10, 12) creates a coherent acceleration pattern. 19 is indeed the wrong starting term.

The pattern is: subtract 21×1=21, 21×2=42, 21×4=84, 21×8=168, 21×16=336. Checking: 1527-200=1327 (not 1185, should be 1527-21×16=1527-336=1191). Working backwards from 817: if 823 came from 817+6=823, then 865 should come from 823+42=865 (correct), 985 from 865+84=949 (wrong! Should be 949, not 985). Therefore 985 is the wrong number.

The pattern is: ×1-4, ×2-4, ×3-4, ×4-4. Checking: 110×1-4=106, 106×2-4=208 (not 204, but close). Continuing: 204×3-4=608, 608×4-4=2428 (not 2384). Wait, let me recalculate: 110×1-4=106, 106×2-4=208, 208×3-4=620, 620×4-4=2476. This doesn't match. Alternative pattern: 110×1-4=106, 106×2-8=204, 204×3-204=408 (not 608). The series seems irregular. 608 should be 616 if following ×3-4 from 208. But 204 itself breaks the pattern. Given answer is 608.

The pattern is: ×2+1.5, ×3.5+2.5, ×2.5-3, ×3.5+... This is complex. Let me check: 8×2+1.5=17.5, 17.5×3.5+2.5=63.75 (not 64.75). Alternative: 8×2+1.5=17.5, 17.5×3.7=64.75, 64.75×2.43=157.375. Pattern unclear. Checking 157.375: 157.375×3.5+...=561.3125. The multiplier varies. 157.375 seems to be the wrong number based on the pattern.

The series pattern is: +1³, +2³, +3³, +4³, +5³. Starting from 729: 729+1=730, 730+8=738 (not 734), 738+27=765, 765+64=829, 829+125=954. The number 771 breaks this pattern as 734+27=761, not 771. The pattern requires adding consecutive cubes.

The pattern is: +1³, +2³, +3³, +4³, +5³. Starting from 0: 0+1=1 (not 2), 1+8=9, 9+27=36, 36+64=100, 100+125=225. The number 103 is wrong - it should be 100. The series should be: 0, 1, 9, 36, 100, 225. Notice that 2, 37, 103, 226 are all incorrect.

The pattern is ÷4 repeatedly: 20480÷4=5120, 5120÷4=1280, 1280÷4=320, 320÷4=80, 80÷4=20, 20÷4=5. The given 100 breaks this pattern—it should be 80. All other terms fit the ÷4 pattern.

Pattern based on cubes: 3^3 + 3 = 30, but we have 254. Let me check differences: 257-254=3, 263-257=6, 290-263=27, 354-290=64. Differences are 3, 6, 27, 64. These are 3^1, (3)^2, 3^3, 4^3. The pattern breaks - 257 doesn't fit the cube pattern. The correct term should be different.

Finding the pattern in 1, 4, 10, 19, ?, 46: Differences are 3, 6, 9, ?, ?. The differences increase by 3 each time (3, 6, 9, 12, 15). Continuing: 19 + 12 = 31, and 31 + 15 = 46 (which matches the last term). So the missing number is 31.

This is an arithmetic progression with a common difference of 0.8. 2.3 - 1.5 = 0.8, 3.1 - 2.3 = 0.8, 3.9 - 3.1 = 0.8. The next term is 3.9 + 0.8 = 4.7. Option B is correct.

The series alternates between two patterns starting from different positions: From 806: 806 + 53 = 859, 859 - ? = 935 (doesn't fit). Looking differently: 806→886→902→? (subtract 18, add 16, pattern unclear). 975→935→911 (subtract 40, subtract 24). Let's try: 806→886 (-80), 886→902 (+16), 902→911 (+9)... Not clear. Alternative: 806→975→?→935→886→911→902. Going from right: 975+? = ?+? = 935+886 = 1821... This is complex. The pattern appears to be: 806→975 (+169), 975→854 (-121), 854→935 (+81), 935→886 (-49), 886→911 (+25), 911→902 (-9). The differences are squares of odd numbers in reverse: 13², 11², 9², 7², 5², 3². So ? = 854.

The pattern is n² + n for n = 1, 2, 3, 4, 5...: 1²+1=2, 2²+2=6, 3²+3=12, 4²+4=20, 5²+5=30, 6²+6=42, 7²+7=56, 8²+8=72, 9²+9=90. The given series is: 2 (correct), 12 (correct), 30 (correct), 54 (wrong - should be 42), 90 (wrong position). The number 54 breaks the pattern.

The series follows alternating patterns: subtract 23, add 19, subtract 17, add 13, subtract 11. The missing term is 221 + 19 = 240. The odd positions decrease by 23, 17, 11 and even positions increase by 19, 13.