Each number is multiplied by 2 to get the next number.

These are the prime numbers (i.e., numbers whose only factors are one and the number itself).

3 - 1 = 2; 6 - 3 = 3; 10 - 6 = 4; 15 - 10 = 5;
The differences between two consecutive numbers are 2, 3, 4, 5, 6, ...
So, 21 - 15 = 6.
Alternatively, 1 + 2 = 3; 1 + 2 + 3 = 6; 1 + 2 + 3 + 4 = 10;
1 + 2 + 3 + 4 + 5 = 15; 1 + 2 + 3 + 4 + 5 + 6 = 21;
The nth term in the sequence is given by n (n + 1)/2. The numbers are often referred to as triangular numbers.

500 - 475 = 25; 475 - 425 = 50; 425 - 350 = 75; 350 - 250 = 100;
The differences between two consecutive numbers are 25, 50, 75, 100, 125, ...
So, 250 - 125 = 125.

200 - 196 = 4; 196 - 180 = 16; 180 - 116 = 64;
The differences between two consecutive numbers are 4, 16, 64, ... (each number starting with 4 is multiplied by 4 to get the next number).
So, 116 - 4(64) = 116 - 256 = -140.

987 - 109 = 878; 878 - 109 = 769; 769 - 109 = 660;
660 - 109 = 551; 551 - 109 = 442;
The difference between two consecutive numbers is 109.

213 + 213 = 426; 426 + 213 = 639; 639 + 213 = 852;
852 + 213 = 1065; 1065 + 213 = 1278;
The numbers simply increase by 213.

The terms are merely the squares of integers starting with 12 in descending order. Thus,
12² = 12 x 12 = 144; 11² = 11 x 11 = 121; 10² = 10 x 10 = 100;
9² = 9 x 9 = 81; 8² = 8 x 8 = 64; 7² = 7 x 7 = 49;
Alternatively, the differences between consecutive terms form the following simple sequence: 23, 21, 19, 17, 15.

The terms are merely the cubes of integers starting with 2. Thus,
2³ = 2 x 2 x 2 = 8; 3³ = 3 x 3 x 3 = 27; 4³ = 4 x 4 x 4 = 64;
5³ = 5 x 5 x 5 = 125; 6³ = 6 x 6 x 6 = 216; 7³ = 7 x 7 x 7 = 343;

The odd terms of the sequence are simply even numbers.
The even terms are 3, 12, 23, 36, ...
12 - 3 = 9; 23 - 12 = 11; 36 - 23 = 13;
The differences between two consecutive even terms are 9, 11, 13, 15, ...
So, 51 - 36 = 15.

The odd terms continually increase by 17, i.e., 3, 20, 37, ...
The even terms also continually increase by 17, i.e., 10, 27, 44, ...
Alternatively, add 7 and 10 alternately starting with 3.

The odd terms continually increase by 9, i.e., 2, 11, 20, ...
The even terms also continually increase by 9, i.e., 7, 16, 25, ...
Alternatively, add 5 and 4 alternately starting with 2.

Each number is one-tenth the number following it.
Thus, (1/10) x 0.99 = 0.099

Each number is 1.5 times the number preceding it.
Thus, 1.5 x 3.75 = 5.625

Each number is two-thirds the number preceding it.
Thus, 72 x 2/3 = 48

The numerators (starting with 1) increase continually by 3, and the denominators (starting with 24) decrease continually by 3.
The numerators are 1, 4, 7, 10, 13, 16, ...
The denominators are 24, 21, 18, 15, 12, 9, ...

6 - 2 = 4; 14 - 6 = 8; 30 - 14 = 16; 62 - 30 = 32; 126 - 62 = 64
The differences between two consecutive numbers are 4, 8, 16, 32, 64, ... (a sequence where each number is multiplied by 2 to get the next number
starting with 4).