I wonder whether there is a way to express a series/sequence (without guessing the pattern), also how to calculate the sum.
For instance, what is the formula (explicit and recursive) for this sequence: 3, 9, 21, 45, 93, 189, 381, …
Well, I do know the answer/explicit formula for this sequence. However, I would really appreciate some help with finding the formula for sequences without having to guess the pattern. With arithmetic and geometric sequences this is easier, but what do we do in this case. Suppose I want to find the formula of the sequence (explicit and recursive) and want to calculate the sum of the 30 first terms.
Help would really be appreciated!
Your sequence $3, 9, 21, 45, 93, 189, 381,\ldots$ has recursive formula $$a_{n+1}=2a_n+3$$ for $n\geqslant 1$ and $a_1=3$.
To obtain an explicit formula, we can do the following trick. We define another sequence $v_n=a_n+3$, $n\geqslant 1$ with first term $v_1=3+3=6$. This is a geometric sequence, indeed:
$$v_{n+1}=a_{n+1}+3=2a_n+3+3=2(v_n-3)+3+3=2v_n$$
so a direct formula of $v_n$ is $v_n=6\cdot2^{n-1}$. Now, $a_n=v_n-3$, hence a direct formula for $a_n$ is given by
$$\boxed{a_n=6\cdot 2^{n-1}-3}$$