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. Tn = Tn+1 + Tn+2 + Tn+3 + ……………………
arn-1 = arn + arn+1 + arn+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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