Different ways are different

Our fifth example is:

$$3, 4, 6, 10, 18, ? %% (5)$$ (9)

Let's rewrite it in the following form:

$$3, (3 + 1), (4 + 2), (6 + 4), (10 + 8), ? %%(5a)$$ (10)

The next is 18 + 16. This would be 34. Now we can reconstruct the rule

$$X_{i} = X_{i-1} + 2^{i} %% (5b)$$ (11)

The answer is right and the rule is right, but there exist a more simple form:

$$X_{i} = X_{i-1} * 2 - 2 %% (5c)$$ (12)

We see that the idea coming first may be not simplest solution. It my be wrong one too. Let's consider the sixth sequence

$$3, 4, 6, ? %%(6)$$ (13)

We can use the solution we already have and write 10. This variant is not the one possible. Other solution is 9:

$$3, (3 + 1), (4 + 2), (6 + 3) %% (6a)$$ (14)

If there exist one solution, one other may exist too. In many cases the consequences may be quite different. This is the problem of a selection.