How do I generalize explicit formulae that have have numbers in them that create an arithmetic sequence?

Question

For example, If I have $$ a_n=\frac{4(3^n)-1}{5}\\ b_n=\frac{6(3^n)-1}{5}\\ c_n=\frac{8(3^n)-1}{5}\\ $$

Clearly $4,6,8 \ldots$ have the explicit formula $2(n+1)$

So how would I generalize these explicit formulae knowing this arithmetic sequence? Is there a mathematical concept or notation that deals with something like this?

Answer 1

You have \begin{eqnarray*} a_n=\frac{4(3^n)-1}{5}\\ b_n=\frac{6(3^n)-1}{5}\\ c_n=\frac{8(3^n)-1}{5}\\ \end{eqnarray*} Lets rewrite them as \begin{eqnarray*} a_{n,1}=\frac{4(3^n)-1}{5}\\ a_{n,2}=\frac{6(3^n)-1}{5}\\ a_{n,3}=\frac{8(3^n)-1}{5}\\ \vdots \end{eqnarray*} & now we can summarise these into one equation \begin{eqnarray*} a_{n,m}=\frac{2(m+1)(3^n)-1}{5}.\\ \end{eqnarray*}

Answer 2

You may use double index for your sequence.

$$ a_{k,n}=\frac{2(k+1)(3^n)-1}{5}$$

That will cover all your terms.