Number Series

The series follows the pattern n^2 + 1: 1^2+1=2, 2^2+1=5, 3^2+1=10, 4^2+1=17, 5^2+1=26, 6^2+1=37. The term 24 is incorrect; it should be 26.

This is a geometric progression with a = 5 and r = 2. The nth term is a * r^(n-1). The 9th term is 5 * 2^(9-1) = 5 * 2^8 = 5 * 256 = 1280.

The series follows the pattern n^3 - 1: 1^3-1=0, 2^3-1=7, 3^3-1=26, 4^3-1=63, 5^3-1=124, 6^3-1=215. The next term is 7^3-1 = 343-1 = 342.

The pattern is: 18 + 6 = 24; 24 + 60 = 84; 84 + 210 = 294; 294 + 504 = 798. The differences are 6, 60, 210, 504. These are 1*2*3, 3*4*5, 5*6*7, 7*8*9. The next difference is 9*10*11 = 990. 798 + 990 = 1788.

The series follows the pattern: 2*1+4=6, 6*4+6=30, 30*8+20=260. Alternatively, the differences are 4, 24, 230, which does not yield a simple pattern; however, the pattern 2*2+2=6, 6*5+0=30, 30*8+20=260 is also inconsistent. Given the options, 3130 is the only logical progression based on (previous term * n) + constant.

The differences are: 73-72=1, 81-73=8 (2^3), 90-81=9 (3^2), 154-90=64 (4^3), 179-154=25 (5^2). The next difference should be 6^3 = 216. 179 + 216 = 395. The next difference should be 7^2 = 49. 395 + 49 = 444.

The series is q, q+3, 2q+3, 2q+6, 3q+6, 3q+9. The pattern for the n-th term is: if n is odd, (n+1)/2 * q + (n-1)/2 * 3; if n is even, n/2 * q + n/2 * 3. For n=6, term = 3q + 9. Since q is the largest negative integer, q = -1. Term = 3(-1) + 9 = 6.

The sequence follows a pattern of alternating operations: 12*4=48, 48+72=120, 120/5=24, 24+12=36, 36/2=18, 18*3=54, 54+72=126, 126/4.6... The pattern is actually three interleaved sequences: (12, 24, 27, ?), (48, 36, 39, 15), (120, 18, 54, 126, 135, 25, 45). A simpler view is groups of three: (12, 48, 120), (24, 36, 18), (54, 126, 27), (39, ?, 15). The missing number is 30.

The terms of the series follow the pattern n^3 - n for consecutive integers starting from n = 2. Specifically, 2^3 - 2 = 6, 3^3 - 3 = 24, 4^3 - 4 = 60, 5^3 - 5 = 120, and 6^3 - 6 = 210. The next term is 7^3 - 7 = 343 - 7 = 336.

The differences are: 17-15=2, 22-17=5, 32-22=10, 49-32=17, 76-49=27. The second differences are: 3, 5, 7, 10. The last difference should be 9 (3, 5, 7, 9). If the last difference were 9, 49+9=58, not 76.

The series repeats every 4 terms (1, 2, 3, 4). The sum of one cycle is 10. 2007 divided by 4 is 501 with a remainder of 3. The sum is 501 * 10 + (1 + 2 + 3) = 5010 + 6 = 5016.

The differences are: ?-6, 21-?, 33-21=12, 48-33=15. The differences are increasing by 3. If the missing term is 12, differences are 6, 9, 12, 15. This fits.