Sequence and Series Class 11 | Ch-8

Sequence And Series 

Geometric Progression

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 a₁, a₂, a₃, a₄, ………a_n, a_n+1  is called geometric progression. 

If   is the common ratio of GP. Where  

General Geometric Progression is  a, ar, ar², ar³, ……….., 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, ar², ar³, ……….., 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 = 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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