### GEOMETRIC PROGRESSION

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__Geometric Progression, Its nth term, Sum to n terms, sum to infinite terms, Properties of G.P.,Special Results__

__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.**

**\[A\:sequence\:a_{1},\: a_{2},\: a_{3},\: a_{4}........,a_{n}, \: a_{n+1}..... is\: called \:geometric\: progression\]\[If \: a_{n+1}=a_{n}\times \: r\: , where \: r \: is\: the\: common \: ratio \: of \: GP \]\[Where\: \: \: r=\frac{a_{n+1}}{a_{n}}\]**

**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**

**\[nth \: term\: in\: GP\: is\: \: ar^{n-1}\]**

**Sum of first n terms in GP is \[S_{n}=\frac{a(r^{n}-1)}{r-1}\: where, r > 1\]**

**Sum of first n terms in GP is\[S_{n}=\frac{a(1-r^{n})}{1-r}\: where,\: \: r < 1\]**

**Sum to infinite terms in GP.**

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

**\[a+ar+ar^{2}+ar^{3}+ar^{4}+........+ ar^{n-1}+ .........\]**

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

**\[S_{\infty }=\frac{a}{1-r}\; , \; Where -1< r< 1 \; or\;\left | r \right | < 1\]**

__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\[1+2+3+4+5......n=\frac{n(n+1)}{2}\]**

**Sum of square of first n natural number is given by\[1^{2}+2^{2}+3^{2}+4^{2}+......+ n^{2}=\frac{n(n+1)(2n+1)}{6}\]**

**Sum of cube of first n natural number is\[1^{3}+2^{3}+3^{3}+4^{3}+......+n^{3}=\left [ \frac{n(n+1)}{2} \right ]^{2}\]**

**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 \[\frac{1}{2},\; \frac{1}{4},\; \frac{1}{16},\; .......,\; \infty\]**

**THANKS FOR YOUR VISIT**

**PLEASE COMMENT BELOW**

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