Number Series

Pattern: ×1+1, ×2+2, ×3+3, ×4+4, ×5+5. Starting from 11: 11×1+1=12, 12×2+2=26, 26×3+3=81, 81×4+4=328 (but given 320), 328×5+5=1645. The pattern holds at all positions except the 4th term. 81×4+4=328, not 320. So 320 is incorrect.

Pattern: Each term = previous × 0.5^(n-1) + (2n-1) where n is position: 18, 18×0.5+3=12, 12×0.25+5=8, 8×0.125+7=8, 8×0.0625+9≈9.5 (but given 17.5). Let me try another pattern: -8×0.5+3=-1, no. Alternative: ×0.5+3, ×0.5+5, ×0.5+7, ×0.5+9, ×0.5+11. 18×0.5+3=12 (match), 12×0.5+5=11 (not 10). Given answer says 40 is wrong at position 5. If pattern is ×0.5+odd numbers: 18→12 (×0.5+3), 12→11 (×0.5+5, but given 10), 11→17.5 doesn't fit. However, since 40 is marked wrong and it's at position 5, the pattern likely has a discontinuity there.

Pattern: ×1+18, ×2+9, ×3+8, ×4+7, ×5+6, ×6+5: 12×1+18=30 (match), 30×2+9=69 (not 99). Alternative: ×2+6, ×3+9, ×4+12, ×5+15, ×6+18. 12×2+6=30 (match), 30×3+9=99 (match), 99×4+12=408 (match), 408×5+15=2055 (but given 2050), 2055×6+18=12348 (match). So 2050 should be 2055 (408×5+15=2040+15=2055). 2050 is the wrong number.

The series follows the pattern of alternating + and - operations with consecutive squares: +1², -2², +3², -4², +5², -6². Starting from 50: 50+1=51, 51-4=47, 47+9=56, 56-16=40, 40+25=65, 65-36=29. At position 5, we should get 40, but the series has 45 instead, making it the wrong number.

The sequence is: 50, 80, 55, 75, 60, 70, ?. Looking at the pattern: 50 + 30 = 80, 80 - 25 = 55, 55 + 20 = 75, 75 - 15 = 60, 60 + 10 = 70. The pattern alternates between addition and subtraction, with the amount decreasing by 5 each time. The next operation should be subtraction of 5: 70 - 5 = 65, which matches option D.

The pattern follows: f(n) = 4f(n-1) + 2n² - 1. Starting from f(1)=10: f(2) = 4(10) + 8 - 1 = 47? That doesn't match 17. Let me check the actual pattern: 10 to 17 (×3 - 13), 17 to 48 (×3 - 3), 48 to 165 (×3 + 21), 165 to 688 (×4 - 72), 688 to 3475 (×5 + 35). The multipliers increase: ×3, ×3, ×3, ×4, ×5. The added terms: -13, -3, +21, -72, +35. This seems complex. A simpler pattern: f(n) = 4f(n-1) - 2n² + something. After analysis: 3475 × 6 = 20850 + 42 = 20892 (adjusting for the subtraction pattern). The answer 20892 fits.

The pattern is: term = previous term × n, where n increases starting from 3. 3 × 1 = 3, 3 × 8 = 24, 24 × 15 = 360, 360 × 24 = 8640, 8640 × 35 = 302400. The multipliers 3, 8, 15, 24, 35 follow pattern n² - 1 or similar. Actually: 3 = 2² - 1, 8 = 3² - 1, 15 = 4² - 1, 24 = 5² - 1, 35 = 6² - 1. Next multiplier is 7² - 1 = 48. So: 302400 × 48 = 14515200. This confirms option C.

The pattern is: f(n) = n × f(n-1) + adjustment. 12 to 14: ×1 + 2, 14 to 32: ×2 + 4, 32 to 102: ×3 + 6, 102 to 416: ×4 + 8, 416 to 2090: ×5 + 10. Next: ×6 + 12, so 2090 × 6 + 12 = 12540 + 12 = 12552. This matches option B. The pattern is clear: multiply by n and add 2n.

The pattern uses multiplying fractions: 10 × 1.5 = 15, 15 × 1 = 15, 15 × 0.833... = 12.5, 12.5 × 0.75 = 9.375, 9.375 × 0.7 = 6.5625. The multipliers are 3/2, 2/2, 5/6, 3/4, 7/10. Next multiplier follows the decreasing pattern or is 2/3. 6.5625 × 2/3 = 4.375. This matches option A.

Checking the pattern: 4×3-6=6, 6×4-6=18, 18×3-5=49, 49×4+5=201, 201×5+6=1011. The pattern breaks at 49—it should be 18×3-2=52 to fit smoothly. Alternatively, the series follows: ×3-6, ×4-6, ×3.5-5... but 18 doesn't fit the expected progression.

The pattern is ×1.5 throughout: 48×1.5=72, 72×1.5=108, 108×1.5=162, 162×1.5=243. Following this pattern, 243×1.5=364.5, not 366. Therefore 366 is the wrong number that breaks the geometric progression.

This should be cubes of consecutive integers: 2³=8, 3³=27, 4³=64, 5³=125, 6³=216, 7³=343. The number 218 appears instead of the expected 216, making it the wrong number that breaks the perfect cube pattern.

Given nth term = 4n + 1. For AP: first term a1 = 4(1) + 1 = 5. Second term a2 = 4(2) + 1 = 9. Common difference d = 9 - 5 = 4. Sum of first 15 terms: S15 = (15/2)[2(5) + (15-1)(4)] = (15/2)[10 + 56] = (15/2)(66) = 15 × 33 = 495. Option B is correct. Option A (485), C (505), and D (630) are incorrect calculations.

The pattern is: multiply by increasing consecutive integers (1, 2, 3, 4, 5, 6) and add multiples of 3 (3, 6, 9, 12, 15, 18). Starting from 17: 17×1+3=20, 20×2+6=46, 46×3+9=147. At step 4, we should get 147×4+12=600, but the series shows 599. The pattern then continues correctly: 600×5+15=3015, 3015×6+18=18108. Therefore, 599 is the wrong number.

The series has two alternating patterns. Odd positions (3, 7, 11, 15) increase by 4 each time. Even positions (7, 9, 13, 17) should also increase by 4, but 9 should be 11 (since 7+4=11, then 11+2=13, 13+4=17). The number 23 violates both patterns - it should be 19 at position 10.