AP vs GP vs HP Comparison

Feature	AP	GP	HP
Definition	$a_{n+1}$ - $a_n$ =d	$a_{n+1}$ / $a_n$ =r	1/ $a_n$ in AP
nth term	a+(n-1)d	ar^(n-1)	No formula
Sum formula	n $\frac{a+l}{2}$	a $\frac{r^n-1}{(r-1)}$	No formula
Mean	$\frac{a+b}{2}$	sqrt(ab)	2 $\frac{ab}{a+b}$
Condition for a,b,c	2b=a+c	$b^2$ =ac	2/b=1/a+1/c
Assume 3 terms	a-d,a,a+d	a/r,a,ar	Convert to AP
Assume 4 terms	a-3d,a-d,a+d,a+3d	a/ $r^3$ ,a/r,ar, $ar^3$	Convert to AP
Infinite sum	Diverges	$\frac{a}{1-r}$ if \|r\|<1	No formula
Log connection	--	log(GP)=AP	--
Exp connection	exp(AP)=GP	--	--

Key relationships: