Number Series

The sequence is 3(1^2), 3(2^2), 3(3^2), which follows the formula 3(n^2). For the 10th term, calculate 3 * (10^2) = 3 * 100 = 300.

The series is 129, 126, 123, 120, 117. This is an arithmetic progression with a common difference of -3. The next term is 117 - 3 = 114.

The series is squares of even numbers: 2^2=4, 6^2=36, 12^2=144, 20^2=400, 30^2=900, 42^2=1764. The bases are 2, 6, 12, 20, 30, 42. Differences of bases: 4, 6, 8, 10, 12. The missing term is 6^2 = 36.

Sequence 1: 11, 16, 21, ..., 211 (a=11, d=5). Sequence 2: 10, 16, 22, ..., 226 (a=10, d=6). Common terms: 16 is the first. Next common term is 16 + LCM(5,6) = 16 + 30 = 46. Terms are 16, 46, 76, 106, 136, 166, 196. Total 7 terms.

a_n = 11 + 4n, A_m = 9 + 5m. 11 + 4n = 9 + 5m -> 5m = 4n + 2. For m to be an integer, 4n+2 must be divisible by 5. n=2 (m=2, value=19), n=7 (m=6, value=39), n=12 (m=10, value=59). n follows arithmetic progression with d=5. Last term < 500: 11 + 4n < 500 -> 4n < 489 -> n <= 122. Terms: 2, 7, 12, ..., 122. Number of terms = (122 - 2)/5 + 1 = 25.

The series follows a pattern of dividing the previous term by 2. 80/2=40, 40/2=20, 20/2=10, 10/2=5, 5/2=2.5.

The series follows a pattern: +5, -3, +5, -3, +5, -3. 5+5=10, 10-3=7, 7+5=12, 12-3=9, 9+5=14, 14-3=11.

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. The next term is 6^2+1 = 37.

The differences between consecutive terms are 1, 3, 6, 10, 15. These differences follow a pattern of adding consecutive integers (1+2=3, 3+3=6, 6+4=10, 10+5=15). Therefore, the next term is 15 + 10 = 25.

The differences between consecutive terms are 3, 6, 9, 12. This is an arithmetic progression of differences, so the next difference should be 15. Adding 15 to 37 gives 52.

The pattern is: 3*2+1=7, 7*2+2=16, 16*2+3=35, 35*2+4=74, 74*2+5=153.

The pattern is alternating subtraction: 36 - 2 = 34, 34 - 4 = 30, 30 - 2 = 28, 28 - 4 = 24. The next step is 24 - 2 = 22.