Number Series

Sequence: 0, -3, -8, -15, -24. Differences: -3, -5, -7, -9. This is a quadratic sequence. General term n^2 - 1 is 0, 3, 8, 15, 24. So the sequence is 1 - n^2. For n=29, 1 - 29^2 = 1 - 841 = -840.

The sequence is 1, 7, 14, 22, 31. The differences are 6, 7, 8, 9. This is a quadratic sequence where the second difference is constant (1). The nth term is given by 0.5n^2 + 4.5n - 4. For n=15, 0.5(225) + 4.5(15) - 4 = 112.5 + 67.5 - 4 = 176.

The sequence is 4, 4.5, 7, 11.5. The differences are 0.5, 2.5, 4.5. This is an arithmetic progression of differences with a common difference of 2. The next differences are 6.5, 8.5. Term 5 = 11.5 + 6.5 = 18. Term 6 = 18 + 8.5 = 26.5.

The sequence is 1, 1.2, 5.4, 13.6. Differences: 0.2, 4.2, 8.2. The differences increase by 4.0. Next difference = 8.2 + 4.0 = 12.2. Next term = 13.6 + 12.2 = 25.8.

Sequence: -1, 8, 23, 44. Differences: 9, 15, 21. Second differences: 6, 6. This is a quadratic sequence: an^2 + bn + c. 2a = 6 => a = 3. 3a + b = 9 => 9 + b = 9 => b = 0. a + b + c = -1 => 3 + 0 + c = -1 => c = -4. Term = 3n^2 - 4. 50th term = 3 * 2500 - 4 = 7500 - 4 = 7496.

Differences: 3, 5, 7. This is an arithmetic progression of differences. The n-th term is a_n = a_1 + sum(2k+1). The sequence follows n^2 + 0.5. For n=26, 26^2 + 0.5 = 676 + 0.5 = 676.5.

The sequence is -9, -6, -1, 6, 15. The differences between consecutive terms are 3, 5, 7, 9. This is an arithmetic progression of differences starting from 3 with a common difference of 2. The nth term is -9 + sum of the first (n-1) terms of the difference sequence, which is -9 + (n-1)(3 + (n-2)). For n=10, term = -9 + 9(3 + 8) = -9 + 99 = 90.

Sequence: -2, 7, 22, 43. Differences: 9, 15, 21. Second differences are constant (6). General form: an^2 + bn + c. 2a = 6 => a=3. 3a+b = 9 => 9+b=9 => b=0. a+b+c = -2 => 3+0+c=-2 => c=-5. Term = 3n^2 - 5. 21st term = 3(441) - 5 = 1323 - 5 = 1318.

The sequence is 3, 8, 15, 24, which follows the pattern n^2 + 2n or (n+1)^2 - 1. For the 24th term, 25^2 - 1 = 625 - 1 = 624.

The sequence is 6, 9, 15, 24. Differences are 3, 6, 9. This is a quadratic sequence. The n-th term is 1.5n^2 - 1.5n + 6. For n=32, 1.5(1024) - 1.5(32) + 6 = 1536 - 48 + 6 = 1494.

The sequence is 2, 6, 12, 20. These are n^2 + n for n=1, 2, 3, 4: 1+1=2, 4+2=6, 9+3=12, 16+4=20.

This is an arithmetic progression with first term a=4 and common difference d=3. The 15th term is a + (n-1)d = 4 + (14 * 3) = 4 + 42 = 46.

The sequence is an arithmetic progression with first term a = 9 and common difference d = 2. The 10th term is a + (10 - 1)d = 9 + 9(2) = 9 + 18 = 27.

The sequence is 1, 3, 6, 10, which are triangular numbers. The n-th term is n(n+1)/2. For n=9, the term is 9(10)/2 = 45.

The sequence is 6, 12, 18, 24, 30, 36. The pattern is multiples of 6. 21 is the wrong number as it should be 24.