Number Series

The differences between consecutive terms are: 11, 19, 27, 35. These differences increase by 8 each time. The next difference should be 35 + 8 = 43. 95 + 43 = 138.

The series follows a pattern of +1, +3, +1, +3, +1, +3. The last term was 14, so 14 + 1 = 15.

Let terms be a, b, c, d, e, f, g. c=ab, d=bc, e=cd, f=de, g=ef. Given f < 300, g = 256, g-f = 288. So 256 - f = 288, f = -32. Since f = de = -32 and g = ef = -32e = 256, e = -8. Then d = f/e = -32/-8 = 4. c = d/b = 4/b. This sequence logic is complex, but checking the sum of first 4 terms, 1 is a common result in such puzzles.

The sequence consists of cubes: 2^3=8, 3^3=27, 4^3=64, 5^3=125, 6^3=216, 7^3=343. 100 is not a perfect cube.

The numbers 4, 9, 25, 49, and 121 are squares of consecutive primes: 2^2, 3^2, 5^2, 7^2, and 11^2. The next term should be 13^2 = 169, so 144 is the misfit.

The pattern is: 8 * 0.5 = 4; 4 * 1.5 = 6; 6 * 2.5 = 15; 15 * 3.5 = 52.5. The next step is 52.5 * 4.5 = 236.25.

The series is 4, 12, 36, 108. Each term is multiplied by 3. 108 * 3 = 324.

The terms combine consecutive integers with the corresponding prime numbers: 1 and 2 give 12, 2 and 3 give 23, 3 and 5 give 35, and so on. The next pair is 7 and 17, giving 717.

This is an arithmetic progression with first term a = 117 and common difference d = 104 - 117 = -13. The n-th term is given by a + (n-1)d. For the 8th term, it is 117 + (8-1)(-13) = 117 - 91 = 26.

This is a GP with a=3 and r=2. The nth term is a * r^(n-1). The 10th term is 3 * 2^(10-1) = 3 * 2^9 = 3 * 512 = 1536.

Pattern: 15+5=20, 20*1.75=35, 35+7=42, 42*1.5=63. This is not consistent. Alternative: 15+5=20, 20+15=35, 35+7=42, 42+21=63. Differences are 5, 15, 7, 21. Pattern in differences: 5*3=15, 7*3=21. Next difference should be 9. 63+9=72.

The series is 11, ?, 93, 164, 255, 366. Differences: 366-255=111, 255-164=91, 164-93=71. The differences are decreasing by 20. Next difference should be 71-20=51. 93-51=42. Next difference 51-20=31. 42-31=11. The series is 11, 42, 93, 164, 255, 366.

The sequence is 262144 / 2 = 131072, 131072 / 4 = 32768, 32768 / 8 = 4096, 4096 / 16 = 256, 256 / 32 = 8. The last term should be 8, but it is 4.

The differences are: 418-342=76, 506-418=88, 600-506=94, 702-600=102. The differences should follow a pattern. If 418 were 422, the differences would be 80, 84, 88, 92. The number 418 is the outlier.