Geometric Progression (G.P.)
A sequence of non-zero numbers is called a geometric progression iff the ratio of any term to its preceding term is constant.
This non-zero constant is usually denoted by $\displaystyle r$ and is called the common ratio.
Thus $\displaystyle a_{1},a_{2},a_{3},...a_{n}$ or $\displaystyle a_{1},a_{2},a_{3},...a_{n}...$ is a G.P. iff $\displaystyle \frac{a_{k+1}}{a_{k}}=r$, for $\displaystyle k\geq 1$.
A finite G.P. can be written as $\displaystyle a,ar,ar^{2},ar^{3},...ar^{n-1}$
An infinite G.P. can be written as $\displaystyle a,ar,ar^{2},ar^{3},...ar^{n-1},...$
Therefore, $\displaystyle n^{th}$ term of G.P. is given by $\displaystyle a_{n}=ar^{n-1}$
Geometric Series
The series $\displaystyle a+ar+ar^{2}+ar^{3}+...+ar^{n-1}$ or $\displaystyle a+ar+ar^{2}+ar^{3}+...+ar^{n-1}+...$ are called finite or infinite geometric series respectively.
Sum of n terms of a G.P.
Let the first term of a G.P. be $\displaystyle a$ and the common ratio be $\displaystyle r$, and $\displaystyle S_{n}$ be the sum of first n terms of G.P., then
$\displaystyle S_{n}=\frac{a(1-r^{n})}{1-r}$
$\displaystyle S_{n}=\frac{a(r^{n}-1)}{r-1}$
Geometric Mean (G.M.)
The geometric mean of two positive numbers a and b is the number $\displaystyle \sqrt{ab}$
Let $\displaystyle G_{1},G_{2},G_{3},...G_{n}$ be n numbers between positive numbers a and b such that $\displaystyle a,G_{1},G_{2},G_{3},...G_{n},b$ is a G.P.
$\displaystyle b=ar^{n+1}$
$\displaystyle r=\left ( \frac{b}{a} \right )^{\frac{1}{n+1}}$
$\displaystyle G_{n}=ar^{n}=a\left ( \frac{b}{a} \right )^{\frac{n}{n+1}}$
Relationship between A.M. and G.M.
Let A and G be A.M. and G.M. of two given positive real numbers a and b respectively. Then,
$\displaystyle A=\frac{a+b}{2}$ and $\displaystyle G=\sqrt{ab}$
$\displaystyle A-G=\frac{(\sqrt{a}-\sqrt{b})^{2}}{2}\geq 0$
Thus, $\displaystyle A \geq G$
A sequence of non-zero numbers is called a geometric progression iff the ratio of any term to its preceding term is constant.
This non-zero constant is usually denoted by $\displaystyle r$ and is called the common ratio.
Thus $\displaystyle a_{1},a_{2},a_{3},...a_{n}$ or $\displaystyle a_{1},a_{2},a_{3},...a_{n}...$ is a G.P. iff $\displaystyle \frac{a_{k+1}}{a_{k}}=r$, for $\displaystyle k\geq 1$.
A finite G.P. can be written as $\displaystyle a,ar,ar^{2},ar^{3},...ar^{n-1}$
An infinite G.P. can be written as $\displaystyle a,ar,ar^{2},ar^{3},...ar^{n-1},...$
Therefore, $\displaystyle n^{th}$ term of G.P. is given by $\displaystyle a_{n}=ar^{n-1}$
Geometric Series
The series $\displaystyle a+ar+ar^{2}+ar^{3}+...+ar^{n-1}$ or $\displaystyle a+ar+ar^{2}+ar^{3}+...+ar^{n-1}+...$ are called finite or infinite geometric series respectively.
Sum of n terms of a G.P.
Let the first term of a G.P. be $\displaystyle a$ and the common ratio be $\displaystyle r$, and $\displaystyle S_{n}$ be the sum of first n terms of G.P., then
$\displaystyle S_{n}=\frac{a(1-r^{n})}{1-r}$
$\displaystyle S_{n}=\frac{a(r^{n}-1)}{r-1}$
Geometric Mean (G.M.)
The geometric mean of two positive numbers a and b is the number $\displaystyle \sqrt{ab}$
Let $\displaystyle G_{1},G_{2},G_{3},...G_{n}$ be n numbers between positive numbers a and b such that $\displaystyle a,G_{1},G_{2},G_{3},...G_{n},b$ is a G.P.
$\displaystyle b=ar^{n+1}$
$\displaystyle r=\left ( \frac{b}{a} \right )^{\frac{1}{n+1}}$
$\displaystyle G_{n}=ar^{n}=a\left ( \frac{b}{a} \right )^{\frac{n}{n+1}}$
Relationship between A.M. and G.M.
Let A and G be A.M. and G.M. of two given positive real numbers a and b respectively. Then,
$\displaystyle A=\frac{a+b}{2}$ and $\displaystyle G=\sqrt{ab}$
$\displaystyle A-G=\frac{(\sqrt{a}-\sqrt{b})^{2}}{2}\geq 0$
Thus, $\displaystyle A \geq G$
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