تصاعد هندسی

${\displaystyle a_n=a{r}^{n-1}}$

${\displaystyle a_n=r{a}_{n-1}}$

${\displaystyle \frac{a_4}{a_3},\frac{a_3}{a_2}}$

${\displaystyle 20,10,5,2.5,1.25,...}$

${\displaystyle \frac{a_4}{a_3}=\frac{2.5}{5}=0.5}$

${\displaystyle \frac{a_3}{a_2}=\frac{5}{10}=0.5}$

${\displaystyle \sum _{k=0}^{n}a{r}^k=a{r}^0+a{r}^1+a{r}^2+a{r}^3+\cdots +a{r}^n.}$

${\displaystyle \sum _{k=0}^{n}a{r}^k=\frac{a(1-r^{n+1})}{1-r}.}$

${\displaystyle \sum _{k=m}^{n}a{r}^k=\frac{a(r^m-r^{n+1})}{1-r}.}$

${\displaystyle \sum _{k=0}^n{k}^s{r}^k.}$

${\displaystyle \frac{d}{dr}\sum _{k=0}^n{r}^k=\sum _{k=1}^{n}k{r}^{k-1}=\frac{1-r^{n+1}}{(1-r{)}^2}-\frac{(n+1)r^n}{1-r}.}$

${\displaystyle (1-r^2)\sum _{k=0}^{n}a{r}^{2k}=a-a{r}^{2n+2}.}$

${\displaystyle \sum _{k=0}^{n}a{r}^{2k}=\frac{a(1-r^{2n+2})}{1-r^2}.}$

${\displaystyle (1-r^2)\sum _{k=0}^{n}a{r}^{2k+1}=ar-a{r}^{2n+3}}$

${\displaystyle \sum _{k=0}^{n}a{r}^{2k+1}=\frac{ar(1-r^{2n+2})}{1-r^2}.}$

${\displaystyle \sum _{k=0}^{\mathrm{\infty }}a{r}^k=\underset{n\to \mathrm{\infty }}{lim}\sum _{k=0}^{n}a{r}^k=\underset{n\to \mathrm{\infty }}{lim}\frac{a(1-r^{n+1})}{1-r}=\underset{n\to \mathrm{\infty }}{lim}\frac{a}{1-r}-\underset{n\to \mathrm{\infty }}{lim}\frac{a{r}^{n+1}}{1-r}}$

${\displaystyle \sum _{k=0}^{\mathrm{\infty }}a{r}^k=\frac{a}{1-r}-0=\frac{a}{1-r}}$

${\displaystyle \sum _{k=0}^{\mathrm{\infty }}a{r}^{2k}=\frac{a}{1-r^2}}$

${\displaystyle \sum _{k=0}^{\mathrm{\infty }}a{r}^{2k+1}=\frac{ar}{1-r^2}}$

${\displaystyle \sum _{k=m}^{\mathrm{\infty }}a{r}^k=\frac{a{r}^m}{1-r}}$

${\displaystyle \frac{d}{dr}\sum _{k=0}^{\mathrm{\infty }}{r}^k=\sum _{k=0}^{\mathrm{\infty }}k{r}^{k-1}=\frac{1}{(1-r{)}^2}}$

${\displaystyle \sum _{k=0}^{\mathrm{\infty }}k{r}^k=\frac{r}{{\left(1-r\right)}^2};\sum _{k=0}^{\mathrm{\infty }}{k}^2{r}^k=\frac{r\left(1+r\right)}{{\left(1-r\right)}^3};\sum _{k=0}^{\mathrm{\infty }}{k}^3{r}^k=\frac{r\left(1+4r+r^2\right)}{{\left(1-r\right)}^4}}$

${\displaystyle \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots =\frac{1/2}{1-(+1/2)}=1.}$

${\displaystyle \frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots =\frac{1/2}{1-(-1/2)}=\frac{1}{3}.}$

${\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{\sin (kx)}{r^k}=\frac{r\sin (x)}{1+r^2-2r\cos (x)}}$

${\displaystyle \sin (kx)=\frac{e^{ikx}-e^{-ikx}}{2i},}$

${\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{\sin (kx)}{r^k}=\frac{1}{2i}\left[\sum _{k=0}^{\mathrm{\infty }}{\left(\frac{e^{ix}}{r}\right)}^k-\sum _{k=0}^{\mathrm{\infty }}{\left(\frac{e^{-ix}}{r}\right)}^k\right]}$

.

${\displaystyle \prod _{i=0}^{n}a{r}^i={\left(\sqrt{a_1\cdot a_{n+1}}\right)}^{n+1}}$

(if ${\displaystyle a,r>0}$

).

${\displaystyle P=a\cdot ar\cdot a{r}^2\cdots a{r}^{n-1}\cdot a{r}^n}$

.

${\displaystyle P=a^{n+1}{r}^{1+2+3+\cdots +(n-1)+(n)}}$

.

${\displaystyle P=a^{n+1}{r}^{\frac{n(n+1)}{2}}}$

.

${\displaystyle P=(a{r}^{\frac{n}{2}}{)}^{n+1}}$

.

${\displaystyle P^2=(a^2{r}^n{)}^{n+1}=(a\cdot a{r}^n{)}^{n+1}}$

.

${\displaystyle 4,x,16}$

${\displaystyle x=\sqrt{4.16}}$

• Hall & Knight, Higher Algebra, p.  ۳۹، ISBN 81-8116-000-2
• Weisstein, Eric W. "Geometric Series". MathWorld.