Number Series

This is an arithmetic sequence with a = -8 and d = 2. The n-th term is a + (n-1)d. The 5th term is -8 + (4)(2) = -8 + 8 = 0.

The sequence is an arithmetic progression with a = 11 and d = 10. The nth term is a + (n-1)d. The 9th term is 11 + (9-1)10 = 11 + 80 = 91.

Arithmetic sequence formula: a_n = a + (n-1)d. Here a = 2, d = 1. a_n = 2 + (n-1)1 = 2 + n - 1 = n + 1.

The sequence is an arithmetic progression with first term a=5 and common difference d=3. The nth term is a + (n-1)d = 5 + (n-1)3 = 3n + 2.

This is an arithmetic progression with a = 10 and d = -3. The 7th term is a + 6d = 10 + 6(-3) = 10 - 18 = -8.

In an AP, the common difference d = second term - first term = third term - second term. 5 - 2x = (6 - x) - 5. 5 - 2x = 1 - x. 4 = x. The terms are 8, 5, 2. The common difference is 5 - 8 = -3.

The sequence is an arithmetic progression with a common difference of 2. The next term is 8 + 2 = 10.

The sequence is an arithmetic progression with first term a = 19 and common difference d = 6. The n-th term is a + (n-1)d = 19 + (n-1)6 = 19 + 6n - 6 = 6n + 13.

This is an arithmetic progression with first term a = 10 and common difference d = -3. The n-th term is given by a + (n - 1)d. For n = 30, the term is 10 + (29 * -3) = 10 - 87 = -77.

This is an arithmetic progression with a=1, d=6. Last term = a + (n-1)d. 223 = 1 + (n-1)6. 222 = (n-1)6. n-1 = 37. n = 38.

This is an arithmetic progression with the first term a = 39 and common difference d = -5. The formula for the n-th term is a + (n-1)d, which is 39 + (n-1)(-5) = 39 - 5n + 5 = -5n + 44.

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

The sequence is 12, 19, 29, 42. The differences are 7, 10, 13, which is an arithmetic progression with d=3. The n-th term is a_n = a_1 + sum of (7 + (i-1)3). For n=20, a_20 = 12 + (19/2)(2*7 + 19*3) = 12 + 9.5(14 + 57) = 12 + 674.5, which suggests a calculation error in the sequence or formula. Re-calculating: a_n = 1.5n^2 + 2.5n + 8. For n=20, 1.5(400) + 2.5(20) + 8 = 600 + 50 + 8 = 658.

Assuming an arithmetic progression: a + 2d = 12, a + 49d = 106. Subtracting gives 47d = 94, so d = 2. Then a = 12 - 4 = 8. The 29th term is a + 28d = 8 + 28(2) = 8 + 56 = 64.