Number Series

This is an arithmetic progression with first term a=1 and common difference d=2. The sum of the first n terms is n^2. For n=10, 10^2 = 100.

The differences between consecutive terms are: 30-23=7, 23-17=6, 17-12=5. Following this pattern, the next difference should be 4. Thus, 12-4 = 8. Checking the next step, 8-3 = 5, which matches the series.

Each term is obtained by multiplying the previous term by 4: 3 x 4 = 12, 12 x 4 = 48, and 48 x 4 = 192. Therefore, the missing term is 192 x 4 = 768.

The series is (2+1) + (4+2) + (8+3) + (16+4) + ... = 3 + 6 + 11 + 20 + ... This is not a standard arithmetic or geometric series. Breaking it into two series: (2+4+8+... 10 terms) + (1+2+3+... 10 terms). Sum = 2*(2^10 - 1)/(2-1) + 10/2 * (1+10) = 2 * 1023 + 5 * 11 = 2046 + 55 = 2101.

The pattern is: 13 * 2 - 1 = 25, 25 * 2 + 1 = 51, 51 * 2 - 1 = 101, 101 * 2 + 1 = 203. The next term is 203 * 2 - 1 = 405.

The series follows a pattern of adding powers of 2: 25+2=27, 27+4=31, 31+8=39, 39+16=55, 55+32=87, 87+64=151. The missing number is 55.

The differences between consecutive terms are: 7-5=2, 11-7=4, 19-11=8. The differences are powers of 2. The next difference should be 16. 19 + 16 = 35. Checking: 35 + 32 = 67, 67 + 64 = 131. The pattern holds.

The series consists of squares of consecutive even numbers: 2^2=4, 4^2=16, 6^2=36, 8^2=64, 10^2=100, and 12^2=144. The missing term is 8^2, which equals 64.

The series follows a pattern of division: 51975 / 5.5 = 9450, 9450 / 4.5 = 2100, 2100 / 3.5 = 600, 600 / 2.5 = 240, 240 / 1.5 = 160, 160 / 0.5 = 320.

The sequence follows a pattern of multiplying by 1.5: 3 * 1.5 = 4.5; 4.5 * 1.5 = 6.75; 6.75 * 1.5 = 10.125; 10.125 * 1.5 = 15.1875.

The series consists of Harshad numbers (numbers divisible by the sum of their digits). The sequence is 12, 18, 20, 21, 24, 27, 30, 36, 40, 42. The 10th number is 42.

Pattern: 6 * 2 = 12, 12 * 4 = 48, 48 * 6 = 288, 288 * 8 = 2304.

The series follows two alternating patterns: the first sequence (20, 19, 17, 14, ?) subtracts 1, 2, 3, 4, resulting in 10. The second sequence (20, 16, 13, 11, ?) subtracts 4, 3, 2, 1, resulting in 10.

The series follows an alternating pattern: +4, -9, +16, -25, +36. Starting from 91, 91 - 25 = 66. Then 66 + 36 = 102, which matches the series.