Number Series

The pattern is an increasing difference of 2 between consecutive terms. The differences are 8, 10, 12, 14, so the next difference is 16. Adding 16 to 54 gives 70. This is a number series where the second-order difference is constant.

The pattern follows n^2 + 3 starting from n=3. 3^2+3=12, 4^2+3=19, 5^2+3=28, 6^2+3=39. The next term is 7^2+3=52. This is a quadratic sequence.

The pattern follows n^2 + 4 starting from n=1. 1^2+4=5, 2^2+4=8, 3^2+4=13, 4^2+4=20, 5^2+4=29. Wait - 10, 17, 26, 37... let me recalculate. Actually: 5=1^2+4, 10=2^2+6, 17=3^2+8, 26=4^2+10, 37=5^2+12. Pattern is n^2 + 2(n+1). For n=6: 36+14=50.

The differences are 12, 20, 30, 42. The second differences are 8, 10, 12. Continuing with second difference of 14, the next first difference is 42+14=56. So the next term is 109+56=165. This is a cubic sequence.

Series pattern: each term = previous term + n³ where n increases. 15 + 1³ = 16 (not 19), but actually 15 + 2² = 19, 19 + 2² + 1 = 24, 24 + 3² = 33, then 33 + 4² = 49 (not 97). The pattern 19 breaks the sequence as it should be based on squares or cubes. 19 is the wrong number.

Pattern: 1(=2¹-1), 5(=2²+1), 13(=2³+5), 31(=2⁴+15), 61(=2⁵+29), 125(=2⁶-3), 253(=2⁷-3). Actually pattern is: term × 2 + 3 gives next. 1×2+3=5, 5×2+3=13, 13×2+5=31, 31×2-1=61 (should be 65), so 61 is wrong. Wait, 31×2-1=61, but pattern suggests 31×2+3=65. However, answer says 31 is wrong. Let's re-verify: 1,5,13,31,61,125,253. Differences: 4,8,18,30,64,128. Pattern of differences: 4×2=8, 8×2+2=18 breaks. Actually series is 2^n - something. If 31 were replaced by 29, pattern would be cleaner. Given answer key, 31 is identified as wrong.

The differences are: 19, 38, 57, 76, 94. The pattern of second differences: 19, 19, 19, 18. The last second difference breaks the pattern - should be 19, making the last term 302+95=397, not 302. So 302 is wrong.

Pattern: 4×1+10=14, 14×4+3=59, 59×5+2=297, 297×6+1=1793 (not 1799), 1793×7+0=12551. 297 fits the pattern but 1799 should be 1793. Alternatively, 297 is correct per pattern.

The pattern is: add consecutive squares starting from 1: 87+1²=88, 88+2²=92, 92+3²=101 (not 102), 101+4²=117, 117+5²=142. So 102 is wrong - should be 101.

Pattern: each term equals (previous term × n) - n where n = 1, 2, 3, 4, 5. Verifying: 5×1-1=4, 4×2-2=6, 6×3-3=15, 15×4-4=56 (not 55), 56×5-5=275. So 55 is wrong; it should be 56.

Pattern: add consecutive prime numbers starting from 2. Verifying: 12+2=14, 14+3=17, 17+5=22, 22+7=29, 29+11=40 (not 41), 40+13=53. So 41 is wrong; it should be 40. The consecutive primes are 2, 3, 5, 7, 11, 13.

Pattern: alternate division and multiplication with powers of 2 (÷2, ×4, ÷8, ×16, ÷32). Verifying: 16÷2=8, 8×4=32, 32÷8=4, 4×16=64, 64÷32=2 (not 3). So 3 is wrong; it should be 2. The operation powers double each time: 2, 4, 8, 16, 32.

Pattern: multiply by an increasing number and add an increasing constant. 2 × 4.5 + 0 = 9; 9 × 3.33 + 3 = 30; 30 × 3.5 + 15 = 105; 105 × 4.15 + 31 = 436 (approx). Following this, 436 × 5.04 + 63 should give about 1909.75, but the series shows 2196, which breaks the pattern.

The pattern is: add 13, then add 26 (double), then add 52 (double again), then add 104 (double again). 7+13=20, 20+26=46, 46+52=98, 98+104=202. Following the pattern, 202+208=410, but the series shows 419, which breaks the established doubling pattern.

Pattern: 237-196=41, 196-155=41, 155-114=41. Each term decreases by 41. Following this pattern: 114-41=73. The next number in the series is 73.