Number Series

The pattern is n^3 + n for consecutive integers: 4^3 + 4 = 64 + 4 = 68 (correct), 5^3 + 5 = 125 + 5 = 130, but the series shows 135 (wrong), 6^3 + 6 = 216 + 6 = 222 (correct), 7^3 + 7 = 343 + 7 = 350 (correct), 8^3 + 8 = 512 + 8 = 520 (correct). The number 135 breaks the pattern and should be 130.

The pattern is increasing differences: 272 - 240 = 32, 306 - 272 = 34, 342 - 306 = 36. The differences increase by 2 each time (32, 34, 36). The next difference should be 38. So the next term is 342 + 38 = 380. Option C (380) is correct.

The series is 22, 24, 28, ?, 52, 84. The differences are: 24-22=2, 28-24=4. These differences double each time: 2, 4, so the next difference should be 8, making the missing number 28+8=36. Continuing: 36+16=52, and 52+32=84, which matches the given numbers. Options A, C, and D don't fit this doubling difference pattern.

The pattern involves adding successive squares: 107-82=25=5², 207-107=100=10², 432-207=225=15², 832-432=400=20². The next difference is 25²=625, so 832+625=1457. Option D matches this result.

The series follows n³-3 where n increases by 1: 5³-3=122, 6³-3=213, 7³-3=340, 8³-3=509, 9³-3=726. Next term is 10³-3=1000-3=997. Option B is correct.

Each term is obtained by dividing the previous term by 2 and subtracting 50: 3420÷2-50=1660, 1660÷2-50=780, 780÷2-50=340, 340÷2-50=120, 120÷2-50=10. Option C (120) fits the pattern.

The pattern is n³+n where n starts at 1: 1³+1=2, 2³+2=10, 3³+3=30, 4³+4=68, 5³+5=130. Next term is 6³+6=216+6=222. Option A matches.

Pattern of differences: 39 - 15 = 24, 74 - 39 = 35, 122 - 74 = 48. Second differences: 35 - 24 = 11, 48 - 35 = 13. The second differences increase by 2 (11, 13, 15, 17). Next second difference = 15, so next first difference = 48 + 15 = 63. Missing term = 122 + 63 = 185. Verification: 265 - 185 = 80, matches pattern.

The pattern is: x0.5, x1, x1.5, x2, x2.5. Starting: 7 x 0.5 = 3.5, not 4. Then 4 x 1 = 4, not 6. If we replace 6 with 4.5: 7 x 0.5 = 3.5, still doesn't work. Testing: 7 x 0.5 = 3.5 (should be 4), 4 x 1 = 4 (should be 6), 6 x 1.5 = 9 ✓, 9 x 2 = 18 (should be 20). If 6 is wrong and should be 5: 7 x 0.5 = 3.5 ✗. If 6 is wrong and should be 4.5: 7, 4, 4.5 doesn't fit. The 6 breaks the pattern.

The pattern is: x2, x3, x4, x5, x6 with subtraction. 3 x 2 = 6, but we have 9. 9 x 3 = 27 ✓, 27 x 4 = 108 - 11 = 97 ✓, 97 x 5 = 485 - 6 = 479 ✓, 479 x 6 = 2874 - 2874 + 7 = 2881 ✓. Wait, checking 3 to 9: 3 x 2 + 3 = 9, 9 x 3 = 27 ✓. But 27 doesn't fit x4-? pattern. Let me reconsider. Actually: 3 x 2 + 3 = 9, 9 x 3 = 27, 27 x 4 - 11 = 97, 97 x 5 - 6 = 479. The multiplier increases, subtraction changes. If 9 is wrong and should be 6: 3, 6, 24, 96... doesn't fit. If 27 is wrong: 3 x 3 = 9, 9 x 3 = 27, should follow x3, x4, x5... The 27 is correct. The 9 breaks the pattern.

Odd positions: 7, -1, -25, -65 with differences -8, -24, -40 (decreasing by 16 each time). Even positions: 8, 24, 58 should have differences +16, +34 (increasing by 18). But 24-8=16, 58-24=34, so 58 is correct. Wait - rechecking: pattern should be 16, 34, 52... 58 fits.

The series should follow the pattern of consecutive odd numbers: 1, 3, 5, 7, 9, 11, 13. The number 10 breaks this pattern as it is an even number. All other numbers are part of the odd number sequence.

The series follows an increasing addition pattern: +2, +3, +5, +4, +6. Starting from 68: 68+2=70, 70+3=73, 73+5=78, 78+4=82, 82+6=88. The number 78 should be checked. Actually, the pattern might be +2, +3, +5, +4, +6 where the additions themselves follow a pattern. If we expect +2, +3, +4, +5, +6, then 78 breaks it (we have +5 instead of +4).

This series shows: 15 ÷ 14 ≈ 1.07, 20 ÷ 15 ≈ 1.33, 39 ÷ 20 = 1.95. The pattern of multiplication factors breaks here. Continuing 20 × 1.33 = 26.6, not 39. The term 39 appears to be incorrect - it should be approximately 26.6-27 to continue the pattern. 39 is the wrong term.

This decreasing series has differences: 786-804=-18, 759-786=-27, 723-759=-36, 678-723=-45, 625-678=-53. The pattern in differences is -18, -27, -36, -45 (each 9 less). But the last difference is -53 instead of -54, breaking the pattern. 625 is incorrect; it should be 623 to maintain the difference of -54.