__GEOMETRIC PROGRESSION__

**A sequence of non-zero numbers is called a geometric progression (G.P.) if the ratio of the term and the term preceding to it is always a constant quantity.**

**A sequence a**_{1}, a_{2}, a_{3}, a_{4},
………a_{n}, a_{n+1} is
called geometric progression.

**If ** is the common ratio of GP. Where
**General Geometric Progression is a**, ar, ar^{2}, ar^{3}, ……….., ar^{n-1}

**First term = a, Second term = ar, and so on and r is the common ratio of GP**

n^{th} term in GP is =
ar^{n-1 }

**Sum of first n terms in GP is **

**Sum of first n terms in GP is**

**Sum to infinite terms in GP.**

**General GP with infinite number of terms is of the following type**

ar, ar^{2}, ar^{3}, ……….., ar^{n-1 }.........∞

**Sum to infinite terms of GP is given by**

__SELECTION OF TERMS IN G.P.__

**\[Three\; terms\; in\; \; G.P.\; are:\; \; \; \frac{a}{r},\; a,\; ar, \; \; here \; common \; ratio\; is\; r\]**

**\[Four\; terms\; in\; \; G.P.\; are:\; \; \; \frac{a}{r^{3}},\; \frac{a}{r},\; ar, \; ar^{3},\;\; here \; common \; ratio\; is\; r^{2}\]****\[Five\; terms\; in\; \; G.P.\; are:\; \; \; \frac{a}{r^{2}},\; \frac{a}{r}, a,\; ar, \; ar^{2},\;\; here \; common \; ratio\; is\; r\]**

__PROPERTIES OF GEOMETRIC PROGRESSION:-__

**1) If all the terms of G.P. be multiplied or divided by the sane non- zero constant, then it remains a G.P. with common ratio r.**

**2) Reciprocal of the terms of a G.P. form a G.P.**

**3) If each term of a G.P. raised to the same power , the resulting sequence also form a G.P.**

**\[4)\; If a_{1},\; a_{2},\; a_{3},........\; a_{n}.... are\; the \; terms \; of \; G.P.,\; \; then\;\\ log\; a_{1},log\; a_{2},.....log\; a_{n}........... \; \;\; \]\[are\; in\; A.P. \; \; and \; \; vice\; -\; versa\]**

__GEOMETRIC MEAN __

**If a, b, c are the three terms of GP then b is said to be the Geometric Mean and is given by****\[b=\sqrt{ac}\: \: or\: \: b^{2}=ac\]**

__Explanation:- __

**If a, b, c are in GP then\[\frac{b}{a} = r = \frac{c}{b}\] ⇒ \[\frac{b}{a} = \frac{c}{b}\]****\[b^{2}=ac\Rightarrow b=\sqrt{ac}\]**

__Componendo and Dividendo__

**If four terms a, b, c, d are proportional then \[\frac{a}{b}=\frac{c}{d}\]**

**When we apply componendo and dividendo then we get \[\frac{a+b}{a-b}=\frac{c+d}{c-d}\]**

**\[or\; \; \frac{Numerator+Denomenator}{Numerator-Denomenator} =\frac{Numerator+ Denomenator}{Numerator-Denomenator}\]**

__SPECIAL RESULTS:-__

**Sum of first n natural number is given by**

**Sum of square of first n natural number is given by**

**Sum of cube of first n natural number is**

**Question:**

The first term of an infinite G.P. is 1 and any term is equal to the sum of all terms that follow it. Find the infinite G.P.

Solution: It is given that: a = 1

**A.T.Q. T**_{n} = T_{n+1} + T_{n+2} + T_{n+3} + ……………………

** ar**^{n-1} = ar^{n} + ar^{n+1} + ar^{n+2} + ……………………

**\[ar^{n-1}=\frac{ar^{n}}{1-r}\]Now cross multiply it and putting a = 1 we get \[r^{n-1}-r^{n}=r^{n}\Rightarrow 2r^{n}=r^{n-1}\]\[\Rightarrow 2r = 1 \Rightarrow r=\frac{1}{2}\]**

**Putting a = 1 and r = 1/2 we get the required sequence as follows \[\frac{1}{2},\; \frac{1}{4},\; \frac{1}{16},\; .......,\; \infty\]**

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