Number Series

The pattern is +1, +4, +1, +4, +1. 3+1=4, 4+4=8, 8+1=9, 9+4=13, 13+1=14, 14+4=18.

The differences are: 30-23=7, 23-17=6, 17-12=5, 12-x=4, x-5=3. So 12-4 = 8.

The series follows the pattern of doubling the previous term and adding 1, or alternatively, the terms are 2^(n+1) - 1. For the last term, 63 * 2 + 1 = 127.

Differences: 21-12=9, 37-21=16, 62-37=25, 98-62=36, 147-98=49. These are squares: 3^2, 4^2, 5^2, 6^2, 7^2. The next difference should be 8^2 = 64. 147 + 64 = 211.

The pattern alternates between subtracting 4 and subtracting 2: 17-4=13, 13-2=11, 11-4=7, 7-2=5, 5-4=1, 1-2=-1.

The pattern is x(n+1) = 2*x(n) - 1. 3*2-1=5, 5*2-1=9, 9*2-1=17, 17*2-1=33, 33*2-1=65.

The differences between consecutive terms are 4, 8, 12. This is an arithmetic progression of differences, so the next difference should be 16. Adding 16 to 40 gives 56.

This is a geometric progression with first term a = 6 and common ratio r = 2. The nth term is a * r^(n-1). 7th term = 6 * 2^(7-1) = 6 * 2^6 = 6 * 64 = 384.

Sum = (n/2) * [2a + (n-1)d]. Here a = -8, d = 2, Sum = 52. 52 = (n/2) * [-16 + (n-1)2] = (n/2) * [2n - 18] = n(n - 9). n^2 - 9n - 52 = 0. Roots are n = 13 and n = -4. Since n must be positive, n = 13.

Arithmetic progression: a=9, d=-4, n=100. Sum = n/2 * (2a + (n-1)d) = 50 * (18 + 99 * -4) = 50 * (18 - 396) = 50 * -378 = -18900.

This is an arithmetic progression with a = 7 and d = 4. Formula: a_n = a + (n-1)d. 403 = 7 + (n-1)4. 396 = (n-1)4. 99 = n-1. n = 100.

The exponents follow the pattern a^n and b^(2n - 1) for the nth term. Substituting n = 6 gives a^6b^11.

AP series: -6, -3, 0... a = -6, d = 3. Sum S_n = (n/2) * [2a + (n-1)d] = 225. (n/2) * [-12 + (n-1)3] = 225. n * (-12 + 3n - 3) = 450. 3n^2 - 15n - 450 = 0. n^2 - 5n - 150 = 0. (n-15)(n+10) = 0. n = 15.

Arithmetic progression with a=8, d=-4, n=50. Sum = (n/2) * [2a + (n-1)d] = (50/2) * [16 + 49(-4)] = 25 * [16 - 196] = 25 * (-180) = -4500.

This is an arithmetic progression with first term a = 93 and common difference d = -3. The 13th term is a + (n-1)d = 93 + (12 * -3) = 93 - 36 = 57.