Number Series

The pattern is adding consecutive even numbers starting from 72: 1480+72=1552, 1552+110=1662 (72+38), 1662+156=1818 (110+46), 1818+210=2028 (156+54). The differences themselves increase by 38, 46, 54 - each increasing by 8. Next difference is 54+8=62, so 2028+272=2300. Alternatively, differences are 72, 110(=72+38), 156(=110+46), 210(=156+54) - the increments 38, 46, 54 increase by 8.

The pattern is adding consecutive even numbers starting from 12: 31+12=43, 43+14=57, 57+16=73, 73+18=91. Each difference increases by 2. The next difference is 20, so 91+20=111. The sequence of differences is 12, 14, 16, 18, 20 - consecutive even numbers. Option A (143) would require an incorrect jump of 52.

The pattern is multiply by 4/3 (or 1.333...): 8748 × 4/3 = 11664, 11664 × 4/3 = 15552, 15552 × 4/3 = 20736, 20736 × 4/3 = 27648. Continuing: 27648 × 4/3 = 36864. Each term is exactly divisible by 3, confirming the pattern.

The pattern is multiply by 3/4 (or 0.75): 40320 × 3/4 = 26880, 26880 × 3/4 = 20160, 20160 × 3/4 = 15120... wait, that doesn't match. Let me check: 16128 × 3/4 = 12096, not 13440. Let me try: 40320, 26880 (×0.67), 20160 (×0.75), 16128 (×0.8), 13440 (×0.833), 11520 (×0.857). Actually, checking: 40320/26880 = 1.5, 26880/20160 = 4/3, 20160/16128 = 1.25, 16128/13440 = 1.2, 13440/11520 = 1.167. Hmm, let me try: 26880 = 40320 × 3/4? No, that's wrong. 26880 × 3/4 = 20160. 20160 × 4/5 = 16128. 16128 × 5/6 = 13440. 13440 × 6/7 = 11520. So the missing term is 20160 (26880 × 3/4).

Pattern: differences are n³ + n² for n=1,2,3,4,5. 1³+1²=2, but actual diff is 12; 2³+2²=12 (matches), 3³+3²=36 (matches), 4³+4²=80 (matches), 5³+5²=150 (matches). For last term, pattern gives 6³+6²=216+36=252, so term should be 305+252=557. But given term is 590, which is 590-305=285 (wrong).

Calculate the pattern of differences: 192-187=5, 216-192=24, 277-216=61. The differences follow the pattern: 5=1×5, 24=2×12, 61=3×20+1. Alternatively, observe that the differences themselves follow a pattern where each difference is obtained by adding consecutive numbers to the previous difference. The next difference is 122, so the missing term is 277+122=399.

The series follows a pattern where the differences themselves increase with a constant second difference. First differences are 528, 575, 624, 675. The second differences are 47, 49, 51 (increasing by 2). The next second difference is 53, making the next first difference 675+53=728. Adding this to 3854 gives 4582.

The pattern is: divide by 4.5, then subtract 2. 945÷4.5=210, 210-2=208; 210÷4.5≈46.67, 46.67-2≈44.67... Wait, let me recalculate. Actually: 945÷4.5=210, 210-2=208 (but we have 52.5). Let me verify: 945÷18=52.5, 52.5-37.5=15. Actually the pattern stated in sub_topic is 'dividing by 4.5 and then subtracting 2'. Let me check: 945÷4.5=210, but next term is 210. So it's just dividing by 4.5. Then 210÷4.5=46.67, but we have 52.5. 210÷4=52.5. 52.5÷3.5=15. 15÷3=5. The divisors are 4.5, 4, 3.5, 3...

The pattern alternates: subtract 4, add 27, subtract 16, add 125, then subtract 64. The subtractions are powers of 2 (2², 4², 8²) and additions follow n³ where n increases by 2 (3³, 5³). 320 - 64 = 284.

This is a complex alternating series. Odd terms increase by 73 each time (4442→4515→4592). Even terms decrease by 75 (3146→3071→2996). However, the answer 2992 doesn't fit the -75 pattern (should be 2996). The series may involve a different pattern.

Pattern: (7.5×8-3)=57, (57×5+10)=295, (295×4-92)=1088✗, should be (295×4-87)=1093. The series follows ×8-3, ×5+10, ×4-87, ×2+2352, ×1+3574. 1088 breaks the ×4 pattern (should be 1093).

Pattern: ×2+2 repeatedly. 456×2+2=914, but given 908✗. Continuing correctly: 914×2+2=1830, 1830×2+2=3662✗(given 10864), 3662×2+2=7326, 7326×2+2=14654. The entire given series after 456 is wrong due to the initial error at 908, making 54312 the wrong number in the context of the intended pattern.

The pattern is: multiply the previous term by 1.5, then increase the multiplier by 0.5 each step. 18 = 12 × 1.5, 45 = 18 × 2.5, 157.5 = 45 × 3.5, and 708.75 = 157.5 × 4.5.

The differences between consecutive terms follow the pattern: 12, 16, 23, 33. Each difference increases by adding consecutive multiples of 3: +4, +7, +10, +13. So the next difference is 33 + 13 = 46, giving 132 + 46 = 178.

The pattern is: divide by 8, then 7, then 6, then 5, then 4. 20160 ÷ 8 = 2520, 2520 ÷ 7 = 360, 360 ÷ 6 = 60, 60 ÷ 5 = 12, 12 ÷ 4 = 3.