Number Series

This is an arithmetic progression with a first term a=2 and a common difference d=4. The n-th term is given by a + (n-1)d, so the 10th term is 2 + (10-1)*4 = 2 + 36 = 38.

This is an arithmetic sequence with a first term a = -4 and a common difference d = -4. The formula for the n-th term is a + (n - 1)d. For the 9th term, this is -4 + (9 - 1)(-4) = -4 + 8(-4) = -4 - 32 = -36.

The sequence is 100, 59, 18. The difference is -41. 100 - 41 = 59, 59 - 41 = 18. The 4th term is 18 - 41 = -23. The 5th term is -23 - 41 = -64.

This is an arithmetic progression with the first term a = 9.9 and common difference d = 6.4 - 9.9 = -3.5. The n-th term is a + (n-1)d. For n = 7, the term is 9.9 + (6 * -3.5) = 9.9 - 21 = -11.1.

The sequence is 0.5, 2.0, 4.5, 8.0. Differences: 1.5, 2.5, 3.5. This is an arithmetic progression of differences. The n-th term is a + sum of (1.5 + (k-1)*1) for k=1 to n-1. a_n = 0.5 + (n-1)/2 * (3 + (n-2)) = 0.5 + (n-1)(n+1)/2 = 0.5 + (n^2-1)/2 = n^2/2. For n=23, a_23 = 23^2 / 2 = 529 / 2 = 264.5.

The listed terms are 1, 2, 4, 7, 11, and 16, with successive differences 1, 2, 3, 4, and 5. The next difference is 6, so the seventh term is 16 + 6 = 22.

The sequence is 2, 8, 18, 32... which can be written as 2(1^2), 2(2^2), 2(3^2), 2(4^2)... The n-th term is 2(n^2). For n=9, the term is 2(9^2) = 2(81) = 162.

The sequence is 10(1^2), 10(2^2), 10(3^2), 10(4^2). The n-th term is 10*n^2. The 8th term is 10*(8^2) = 10*64 = 640.

The sequence is 0, 3, 8, 15... which is n^2 - 1 for n=1, 2, 3, 4. The 11th term is 11^2 - 1 = 121 - 1 = 120.

The sequence is 2, 5, 10, 17... which is n^2 + 1 for n=1, 2, 3, 4. The 12th term is 12^2 + 1 = 144 + 1 = 145.

Sequence: 3(1^2), 3(2^2), 3(3^2), 3(4^2)... The n-th term is 3*n^2. For n=18, 3*(18^2) = 3 * 324 = 972.

Sequence: -1, 2, 7, 14. Differences: 3, 5, 7. This is an arithmetic progression of differences. The n-th term is n^2 - 2. Check: 1^2-2 = -1, 2^2-2 = 2, 3^2-2 = 7, 4^2-2 = 14. 12th term = 12^2 - 2 = 144 - 2 = 142.

The sequence is 0, 3, 8, 15... which is n^2 - 1 for n = 1, 2, 3, 4. The 24th term is 24^2 - 1 = 576 - 1 = 575.