sequences and series - How Can I Represent These Progressions in Sigma Notation?

I would like to represent the following finite progressions in sigma notation:

1. $Finding\ the \ n^{th} \ term \ of \ a \ geometric \ progression$: $a_n=a_1(r^{n-1})$, where $a_1$ is the first time and $r$ is the common ratio




2. $The \ sum \ of \ a \ geometric \ progression: \ S_n=a_1\frac{1-r^n}{1-r}$



3. $Determining \ the \ n^{th} \ term \ of \ an \ arithmetic \ progression: a_n=a_1+(n-1)d$,
   $\ $ where $d$ is the common difference


4. And finally, the sum of an arithmetic progression: $S_n=\frac{n}{2}(2a_1+(n-1)d)$

Answer

You wish to expression the sum of the first $n$ terms of an geometric progression and the sum of the first $n$ terms of an arithmetic progression in summation notation.

Sum of a geometric progression: If the initial term is $a_1$ and the common ratio is $r$, then the $k$th term of the geometric progression is $a_k = a_1r^{k - 1}$. Hence, the sum of the first $n$ terms of the geometric progression is
$$S_n = \sum_{k = 1}^{n} a_1r^{k - 1} = \begin{cases} a_1 \dfrac{1 - r^n}{1 - r} & \text{if $r \neq 1$}\\ na_1 & \text{if $r = 1$} \end{cases}$$
Notice that the index of the variable must be different from the index of the upper limit. Otherwise, all $n$ terms in the sum would be equal to $a_1r^{n - 1}$.

Sum of an arithmetic progression: If the initial term is $a_1$ and the common difference is $d$, then the $k$th term of the arithmetic progression is $a_k = a_1 + (k - 1)d$. Hence, the sum of the first $n$ terms of the arithmetic progression is

$$S_n = \sum_{k = 1}^{n} [a_1 + (k - 1)d] = \frac{n}{2}[2a_1 + (n - 1)d] = \frac{n(a_1 + a_n)}{2}$$