### Common Errors in Secondary Mathematics

Common Errors Committed  by the  Students  in Secondary Mathematics   Errors  that students often make in doing secondary mathematics  during their practice and during the examinations  and their remedial measures are well explained here stp by step.  Some Common Errors in Mathematics

### 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, ………an, an+1  is called geometric progression.
If   is the common ratio of GP. Where
General Geometric Progression is  a, ar, ar2, ar3, ……….., arn-1

First term = a, Second term = ar,  and so on and  r is the common ratio of GP
nth term in GP is =  arn-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, ……….., arn-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.        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 $\frac{1}{2},\; \frac{1}{4},\; \frac{1}{16},\; .......,\; \infty$