Number Series

Pattern: ×7.5, ×6.5, ×5.5, ×4.5 should give 26, 195, 1267.5, 6971.25, 31370.625. Actual has 6972.25 instead of 6971.25 - an error of 1.

The pattern is: multiply by 1 and add 1, multiply by 2 and add 4, multiply by 3 and add 8, multiply by 4 and add 12. So: 15×1+1=16, 16×2+4=36, 36×3+8=116 (but should be 116), 116×4+12=476 (not 484). Actually, rechecking: 15×1+1=16, 16×2+4=36, 36×3+8=116, 116×4+12=476. So 484 is wrong. But the question says 116 is correct. Let me verify another pattern: ×1+1, ×2+4, ×3+8, ×4+12 works correctly with 476, not 484. Wait, let me check: 36×3+8=116, then 116×4+12=476. So if 484 is there instead of 476, then 484 is wrong. But the answer says 116 is wrong.

The series follows the pattern: square then +1, then square then -1, alternating. 8²+1=65 (not 165), 9²-1=80, 10²+1=101, 11²-1=120, 12²+1=145. The series should be 65, 80, 101, 120, 145, 168. The term 165 breaks this alternating pattern - it should be 65 (8²+1).

The series follows: x1-1, x2-2, x3-3, x4-4, x5-5. Starting from 75: 75×1-1=74, 74×2-2=146, 146×3-3=435 (not 430), 435×4-4=1736, 1736×5-5=8675. The term 430 is wrong - it should be 435. Option B correctly identifies this.

The pattern is adding 3 to each number: 13 + 3 = 16, 16 + 3 = 19, 19 + 3 = 22, 22 + 3 = 25. However, 24 appears instead of 25, and 28 = 25 + 3. Therefore, 24 is the wrong number.

The differences between consecutive terms are: 5-2=3, 11-5=6, 20-11=9, 32-20=12, 47-32=15. The differences increase by 3 each time. The next difference should be 18, so missing term = 47 + 18 = 65. Verifying: 65 + 15 = 80 (pattern holds). Options A, C, D don't fit.

Analyze the differences between consecutive terms: 6-3=3, 10-6=4, 16-10=6, 24-16=8. The pattern shows differences increasing by 1, then 2, then 2. However, a clearer pattern is: add 3, add 4, add 6, add 8, add 12. Each addition increases by 1, 2, 2, 4... Alternatively, this could be n+3 where n is prime-related. The pattern 3, 4, 6, 8, 12 suggests doubling prime increments. The next term is 24 + 12 = 36.

Calculate the differences between consecutive terms: 23-10=13, 43-23=20, 74-43=31, 120-74=46. Now analyze the second differences: 20-13=7, 31-20=11, 46-31=15. The second differences increase by 4 each time (7, 11, 15). The next second difference should be 15+4=19, so the next first difference is 46+19=65. Therefore, the next term is 120+65=185.

The series alternates between a number and its cube: 2^3 = 8, 3^3 = 27, 4^3 = 64. Following this pattern, 5^3 should be 125, but the series shows 225 instead (which is 15^2). Therefore, 225 is the wrong number that breaks the established cube pattern.

In the series 15, 72, 363, 1815, 9063, 45312, 226563, the pattern is: multiply by 5, then subtract 12. Checking: 15×5-12=63 (not 72). 72×5-12=348 (not 363). 363×5-12=1803 (not 1815). The pattern is multiply by 5, then subtract 12, but 1815 doesn't fit correctly. Let's recalculate: 363×5=1815, 1815-12=1803. So 1815 should be 1803. Or if 1815 is correct, then the rule is simply ×5: 15×5=75 (not 72). The pattern is inconsistent. Checking: 15×5-3=72, 72×5-? 72×5=360, 363=360+3. The pattern seems to be ×5 then alternating -3 and +3. 15×5-3=72, 72×5+3=363, 363×5-3=1812 (not 1815). So 1815 is wrong.

In the series 88, 44, 44, 66, 130, 330, 990, the pattern is: divide by 2, then multiply by 1, 1.5, 2, 2.5, 3, 3.5. Checking: 88÷2=44, 44×1=44 (correct), 44÷2=22 (not 66). Let's try: 88÷2=44, 44×1=44, 44×1.5=66 (correct), 66÷2=33, 33×2=66 (not 130). Another pattern: 88÷2=44, 44×1=44, 44+22=66, 66+64=130, 130+200=330, 330+660=990. The additions double each time: 22, 22, 64, 200, 660. This doesn't work. Testing 130: if 66×2-2=130, then 130×2.5=325 (not 330). 130 breaks the pattern.

The series follows the pattern +5, -10, +15, -20, +25, ... Starting from 104: 104+5=109, 109-10=99, 99+15=114, 114-20=94. The next term should be 94+25=119, but we have 120 instead. Therefore, 120 is the wrong term that breaks the alternating pattern.

In the series 2, 3, 4, 11, 48, 389, the number 3 doesn't fit the underlying pattern. A possible pattern with 3 removed: 2^2 - 0 = 2 (or starting point), 4^2 - 5 = 11, 11^2 - 73 = 48, etc. Another interpretation: the series follows n x previous - k pattern where 3 breaks the sequence. The key is recognizing that 3 is the outlier.

The pattern is n^3 + n starting from n=3: 3^3 + 3 = 30, 4^3 + 4 = 68, 5^3 + 5 = 130. Following this pattern, 6^3 + 6 should be 222, but the series has 224. The remaining terms 7^3 + 7 = 350 and 8^3 + 8 = 520 fit the pattern correctly.

Pattern is multiply by 0.4 each time: 980 × 0.4 = 392, 392 × 0.4 = 156.8, 156.8 × 0.4 = 62.72, but the series has 62.78. The term 62.78 is wrong; it should be 62.72. Continuing: 62.72 × 0.4 = 25.088, and 25.088 × 0.4 = 10.0352.