Sequence Progression

Geometric Progression

• A geometric progression is a sequence of the form $a,ar,a{r}^2,\dots ,a{r}^n,\dots$ where the initial term $a$ and the common ratio $r$ are real numbers.
• It is a discrete analogue of the exponential function $f(x)a{r}^x$.

Example

Find the formula of the following geometric progression if we start at $n=0$:

$1,-1,1,-1,1,\dots$

$b_n=(1{)}^n$

Arithmetic Progression

• An arithmetic progression is a sequence of the form $a,a+d,a+2d,a+nd,\dots$ where the initial term $a$ and the common difference $d$ are real numbers.
• It is a discrete analogue of the linear function $f(x)=dx+a$.

Example

Find the formula of the following arithmetic progression if we start at $n=0$:

$-1,3,7,11,\dots$

$s_n=4n-1$