تصاعد هندسی

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

${\displaystyle a_{n}=r\,a_{n-1}}$

${\displaystyle {\frac {a_{4}}{a_{3}}},{\frac {a_{3}}{a_{2}}}}$

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

${\displaystyle {\frac {a_{4}}{a_{3}}}={\frac {2.5}{5}}=0.5}$

${\displaystyle {\frac {a_{3}}{a_{2}}}={\frac {5}{10}}=0.5}$

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

${\displaystyle {\begin{aligned}(1-r)\sum _{k=0}^{n}ar^{k}&=(1-r)(ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots +ar^{n})\\&=ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots +ar^{n}\\&{\color {White}{}=ar^{0}}-ar^{1}-ar^{2}-ar^{3}-\cdots -ar^{n}-ar^{n+1}\\&=a-ar^{n+1}\end{aligned}}}$

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

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

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

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

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

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

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

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

${\displaystyle \prod _{i=0}^{n}ar^{i}=\left({\sqrt {a_{1}\cdot a_{n+1}}}\right)^{n+1}}$ (if ${\displaystyle a,r>0}$).

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

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

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

${\displaystyle P=(ar^{\frac {n}{2}})^{n+1}}$.

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

${\displaystyle 4,x,16}$

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

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