The simple code

Let's begin. Our first sequence is trivial.

$$1, 2, 3, 4, 5, 6, ?$$ (1)

The answer is of course 7. To find it you need only to read this sequence. The solution comes "automatically" by reading.

The second example is a bit more complex:

$$1, 8, 27, 64, ? %%(2)$$ (2)

This is the sequence of cubes. The answer is 125, 5 cube. I mean you have found the answer no such quick as the first. The complexity is not in calculation, but in searching the rule. Let's write the rules of our examples:

$$X_{i} = X_{i-1} + 1$$ (3)

$$\setcounter{exampleNum}{2}\setcounter{exampleSubNum}{2}X_{i} = ( 1 + i )^{3}$$ (4)

Here and below "i" is a natural number $X_i$ is the i-th element. The 0th element is the first number of the sequence. It may and may be calculated by the formula. $x^y$ is the y-th power of x.