Number Series

Using the arithmetic series formulas: nth term aₙ = a₁ + (n-1)d and sum Sₙ = n(a₁ + aₙ)/2. Given a₃ = 114, aₙ = -27, and S = 2325. From a₃ = a₁ + 2d = 114, we get a₁ = 114 - 2d. From S = n(a₁ + aₙ)/2 = n((114-2d) + (-27))/2 = n(87 - 2d)/2 = 2325. From aₙ = a₁ + (n-1)d: -27 = (114-2d) + (n-1)d = 114 - 2d + nd - d = 114 + nd - 3d. Solving gives d = -3, n = 60, a₁ = 120. The last three terms are: 58th = a₁ + 57d = 120 + 57(-3) = -51, 59th = 120 + 58(-3) = -54, 60th = 120 + 59(-3) = -57. Wait - this doesn't match option D. Let me recalculate: -27 = a₁ + (n-1)(-3), so a₁ = -27 + 3(n-1). Also a₁ = 114 - 2(-3) = 120. So 120 = -27 + 3n - 3, so 3n = 150, n = 50. Last three: 48th = 120 + 47(-3) = -21, 49th = 120 + 48(-3) = -24, 50th = 120 + 49(-3) = -27. This matches option D.

The pattern is (n * 2) + 1, then (n * 2) + 2, then (n * 2) + 3, etc. 4*2+1=9; 9*2+2=20; 20*2+3=43; 43*2+4=90. The next step is 90*2+5 = 185. 170 and 222 do not follow this incremental multiplier-addition rule.

The pattern is +4, +8, +12, +16, +20. Starting from 7: 7+4=11, 11+8=19, 19+12=31, 31+16=47, 47+20=67. Each increment increases by 4. The missing number is 47, which fits the pattern between 31 and 67.

The pattern is +4, +8, +12, +16, +20. Starting from 3: 3+4=7, 7+8=15, 15+12=27, 27+16=43, 43+20=63. The differences form an arithmetic progression. The missing number is 15.

The pattern adds consecutive prime numbers: 9+2=11, 11+4=15, 15+6=21, 21+8=29, 29+10=39. The differences are 2, 4, 6, 8, 10 - consecutive even numbers. The missing number is 11.

To solve this question, the user needs to identify the pattern in the given series of numbers.

The first number is 9, and each subsequent number is obtained by multiplying the previous number by a multiple of 2.

9 x 2 = 18 18 x 4 = 72 72 x 6 = 432 432 x 8 = 3456

Therefore, the missing number in the series is obtained by multiplying 432 by 8, which equals 3456.

So, the Answer is: C. 3456

This is an arithmetic progression with a common difference of 13. Each term increases by 13: 13+13=26, 26+13=39, 39+13=52, 52+13=65. The missing number is 52.

The pattern follows subtracting consecutive integers: 124-19=105, 105-18=87, but then 87-17=70, not 75. The number 87 breaks the pattern; it should be 88 to maintain the sequence (124-19=105, 105-18=88, 88-17=71, 71-16=55, 55-15=40) or 75 should be 70.

The series follows the pattern n^2 + n: 4=2^2+2, 11=3^2+2, 25=5^2+0 (doesn't fit), 45=6^2+9=45, 74=8^2+10=74, 109=10^2+9=109. The differences are: 11-4=7, 25-11=14, 45-25=20, 74-45=29, 109-74=35. Second differences: 14-7=7, 20-14=6, 29-20=9, 35-29=6. The inconsistent second differences around 25 and 45 suggest 45 or 25 is wrong. The answer key says 45 is wrong.

The pattern alternates between adding and subtracting 23. Starting with 582: 582+23=605, 605-23=582 (not 588), 582+23=605, 605+23=628 (not 611). The correct pattern should be: 582, 605, 582, 605, 628, 651, 674 or similar. The number 634 appears where 628 should be if adding 23 consistently.

92 is the wrong number. The pattern is: divide by 6 and add 12 repeatedly. 2880/6+12=480, 480/6+12=92, but 92/6+12=27.33 (not 24). The pattern requires 92 to be 80 (80/6+12=25.33, still not 24). The correct pattern seems to be: divide by 6, 5, 4, 3, 2 respectively: 2880/6+12=492 (not 480). The number 92 breaks whatever pattern is intended.

The pattern is: (15*1)+1=16, (16*2)+2=34, (34*3)+3=105, (105*4)+4=424, (424*5)+5=2125. The sequence lists 2124 instead of 2125, making 2124 the wrong number.

The pattern is: each term × 4 + alternating increment (13, 5, 5, 13, 13). 9×4+13=49, 49×4+5=201, 201×4+5=1009, 1009×4+13=4041, 4041×4+13=20209, 20209×4+13=80841.

First sequence: 17,21,25,... (difference 4). Terms = 17 + 4(n-1) for n=1 to 100. Second sequence: 16,21,26,... (difference 5). Terms = 16 + 5(n-1) for n=1 to 91. Common terms satisfy 17 + 4a = 16 + 5b → 4a - 5b = -1. Solving: terms are 21,41,61,...,401 (21 + 20k for k=0 to 19). Total = 20 terms.

The sequence 0, 1, 10, 11 represents binary (base-2) counting: 0₂=0, 1₂=1, 10₂=2, 11₂=3. The next number in binary after 11 is 100, which equals 4 in decimal. This is a pattern recognition question testing understanding of number systems.