Logarithms: Sequences and Series

Logarithms: & Law of logarithms

If $b^x$= N then x is called logarithm of N to the base b, it is written as x= ${\log }_b$N
1. Logarithm of unity to any base is zero ie ${\log }_a$1= 0
2. Logarithm of a number to the same base is one ie ${\log }_a$a= 1
3. Product rule ${\log }_a$MN= ${\log }_a$M + ${\log }_a$N
4. Quotient rule ${\log }_a$$\frac{M}{N}$ = ${\log }_a$M - ${\log }_a$N
5. Power rule ${\log }_a$$M^n$ = n ${\log }_a$M
6. Base changing rule: ${\log }_a$M = ${\log }_b$ M.${\log }_a$b

Note 1: ${\log }_b$a.${\log }_a$b=1

Note 2: ${\log }_b$a= $\frac{1}{{\log }_a^b}$

Note 3: $\frac{{\log }_b^M}{{\log }_b^a}$= ${\log }_a^M$
7. System of logarithms with base 10 is called Common logarithms
8. System of logarithms with base e is called Natural logarithms

Note:- Usually in Natural logarithms, we do not write the base e.

Arithmetic Progresion

A sequence of the form a, a+d, a+2d, a+3d............ is called an AP if the difference of a term and the previous term is always constant.

The constant difference is called the common difference. The first term of an A.P is usually denoted by 'a' or 't', $n^{\mathrm{th}}$
term is denoted by '$a_n$' or '$t_n$' and the common difference is denoted by 'd'.

a. $n^{\mathrm{th}}$ term, $a_n$ = a+(n-1)d. and n= $\frac{a_n-a_1}{d}$+1

b. Sum of n terms, $S_n$= $\frac{n}{2}$[2a + (n-1)d] = $\frac{n}{2}$[$1^{\mathrm{st}}$term + $n^{\mathrm{th}}$ term]

c. If a,x,b are in A.P. Then x is called the arithmetic mean between a and b. and x= Am= $\frac{a=b}{2}$

d. If a, $x_1$, $x_2$.....$x_n$, b are in A.P. then $x_1$, $x_2$, .... $x_n$ are called then n AM.'s between a and b Here $x_1$= a+d;

$x_2$= a+2d ................ Where d= $\frac{b-a}{n+1}$

e. Sum of n Am's between a & b= $\frac{n}{2}$(a+b)

f. If 3 numbers a, b, c are in A.P. then 2b= a+c


Arithmetic Mean (A.M)
If a, b, c are in A.P then b= $\frac{a+c}{2}$

(i) If a, $x_1$, $x_2$, $x_3\mathrm{..........}{x}_n,$b are in A.P then $x_1$, $x_2$, ........$x_n$ are called the n, A.M.s

between a and b.

(ii) Sum of n A.M.s, $x_1$, $x_2$+.............+$x_n$= n$\left[\frac{a+b}{2}\right]$

(iii) $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ is the A.M between a and b if n= 0.

(iv) Any three numbers in A.P can be taken as a-d, a, a+d.

(v) Any four numbers in A.P can be taken as a - 3d, a-d, a+d, a+3d.

(vi) Any five numbers in A.P can be taken as a - 2d, a-d, a, a+d, a+2d.

Geometric Progression :

A Sequence of the form a, ar, ${\mathrm{ar}}^2$ , ${\mathrm{ar}}^3$.................... is called a GP. A sequence in which each term is the same multiple of the preceeding term is called Geometric Progression. The constant multiple is known as the common ratio. The common ratio of a G.P is denoted by 'r'

a. $n^{\mathrm{th}}$ term, $a_n$= ${a\mathrm{.}r}^{n-1}$

b. Sum of n terms $S_n$ = a $\frac{\left({1-r}^n\right)}{1-r}$= $\frac{a(r^n-1)}{r-1}$

c. If |r| $<$1, the sum of infinite terms $S_{\infty }$= $\frac{a}{1-r}$

d. If a, x, b are in GP then x is called the G.M. between a and b and x= GM= $\sqrt{\mathrm{ab}}$

e. If a, $x_1$, $x_2$, $x_3$,......... $x_n$, b are in G.P, then $x_1$, $x_2$, $x_n$ are called the n G.M's between a and b. Here $x_1$=

ar, $x_2$= ${\mathrm{ar}}^2$, $x_3$= ${\mathrm{ar}}^3$,.................. where r= ${(b/a)}^{1/n+1}$

f. Product of n GM's between a & b= ${\left(\mathrm{ab}\right)}^{n/2}$

g. If a, b, c are in G.P then $b^2$ = ac


Shortcut method for recurring decimals

(i) 0.625 = $\frac{625-6}{990}$= $\frac{619}{990}$

(ii) 0.358= $\frac{358-3}{990}$= $\frac{355}{990}$

(iii) 1.423= 1+ $\frac{423-4}{990}$= $\frac{1409}{990}$

Method

(i) The Numerator of the vulgar fraction is obtained by substracting the non-recurring

figure from the given figure.

(ii) The denominator consists of as many 9's as there are recurring figures and as many

zero as there are non-recurring figures.

(iii) Three numbers in G.P can be taken as $\frac{a}{r}$, a, ar.

(iv) Four numbers in G.P can be taken as $\frac{a}{r^3},\frac{a}{r},\mathrm{ar},{\mathrm{ar}}^3\mathrm{.}$

(v) Five numbers in G.P can be taken as $\frac{a}{r^2},\frac{a}{r},a,\mathrm{ar},{\mathrm{ar}}^2$


Harmonic Progression:

A Sequence is said to be in harmonic progression if the sequence formed by the reciprocals of its term is in AP ie, If $a_1$, $a_2$, $a_3$,................ $a_n$ are in HP. Then $\frac{1}{a_1}$, $\frac{1}{a_2}$, $\frac{1}{a_3}$, ................ $\frac{1}{a_n}$ are in A.P.
a. To find $n^{\mathrm{th}}$ term of a HP, find the $n^{\mathrm{th}}$ term of the corresponding AP (say x) & then $n^{\mathrm{th}}$ term of the HP= 1/x
b. If a, x, b are in H.P, then x is called the Harmonic mean between a and b and x= H.M.= $\frac{2\mathrm{ab}}{a+b}$
c. If a, $x_1$, $x_2$..... $x_n$, b are in HP, then $x_1$, $x_2$, ....$x_n$ are called then n HM.'s between a and b
d. If a, b, c are in H.P then b= $\frac{2\mathrm{ac}}{a+c}$
e. 2,3,6................ is an example of HP
f. The series k,k,k................... is in AP, GP, & HP

Relation between AM, GM & H.M.

If A, G, H are the A.M, G.M. and H.M between two positive numbers a and b, then A, G, H form a decreasing GP. ie A$\ge$G$\ge$H and $G^2$= AH where A= $\frac{a=b}{2}$, G= $\sqrt{\mathrm{ab}}$; H= $\frac{2\mathrm{ab}}{a+b}$

(A) Arithmetico- Geometric Progression (A.G.P)

A sequence of the form a, (a+d)r, (a+2d)$r^2$, (a+3d)$r^3$............. is called an A.G.P
a. $n^{\mathrm{th}}$ term, $a_n$= [a+ (n-1)d].$r^{n-1}$
b. Sum to n terms, $S_n$= $\frac{a}{(1-r)}$+ $\frac{\mathrm{dr}(1-r^{n-1})}{{(1-r)}^2}$- $\frac{[a+(n-1)d]r^n}{(1-r)}$
c. If |r| $<$1, sum to infinite terms $S_{\infty }$= $\frac{a}{1-r}$+$\frac{\mathrm{dr}}{{(1-r)}^2}$

(B) Sum of powers of Natural Numbers

a. Sum of first n natural numbers 1+2+3+4+............+n= $\mathrm{\Sigma }$n= $\frac{n(n+1)}{2}$

b. Sum of squares of first n natural numbers $1^2$+$2^2$+$3^3$+..$n^2$= $\mathrm{\Sigma }$ $n^2$= $\frac{n(n+1)(2n+1)}{6}$

c. Sum of cubes of first n natural numbers $1^3$+ $2^3$+.......$n^3$=$\mathrm{\Sigma }$ $n^3$=${\left[\frac{n(n+1)}{2}\right]}^2$

(C) If the terms a progression are in G.P then logarithm of the terms are in A.P.
(D) A sequence of an A.P if the sum of its n term is of the form ${\mathrm{An}}^2$+ Bn. Where A & B are constants.