# Algebra

Sequence

A sequence is an ordered set of numbers. Some sequences are simply random values; other sequences have a definite pattern,a1 , a2 ,a3…. anis the infinite sequence.Eg3, 6,9,12is the finite sequence.

Types of sequences

Arithmetic Sequence

Geometric Sequence

Harmonic Sequence

Series

The sum of the terms of an infinite sequence is called series. The sum of a finite sequence has defined first and last terms, whereas a series continues indefinitely.

If a1 , a2 ,a3…. an…is the infinite sequence.

Then, Sn= a1 + a2 +a3+…annk= 1is the infinite series

Arithmetic series:.

The nth term of the arithmetic series is given bytn = a + (n – 1)d

l = a + (n – 1)d

1. Arithmetic mean (A.M) = $\frac{{{\rm{a}} + {\rm{b}}}}{2}$

2.If m1, m2, m3,….,mn are arithmetic means then;

1 = a + d

m2 = a + 2d

3 = a + 3d

………………..

mn = a + nd, where d = $\frac{{{\rm{b}} - {\rm{a}}}}{{{\rm{n}} + 1}}$

3. S= $\frac{{\rm{n}}}{2}$ [2a +(n – 1)d]or Sn­ – $\frac{{\rm{n}}}{2}$(n + 1)

Geometric series

1.tn­ = arn-1

2. Geometric mean (G.M) = $\sqrt {{\rm{ab}}}$

3. If g1, g2, g3,…..gn are geometric means then,

Gn = arn, where r = ${\left( {\frac{{\rm{b}}}{{\rm{a}}}} \right)^{\frac{1}{{{\rm{n}} + 1}}}}$

4. Sn =$\frac{{{\rm{a}}\left( {{{\rm{r}}^{\rm{n}}} - 1} \right)}}{{{\rm{r}} - 1}},{\rm{r}} \ne 1$ or, Sn = $\frac{{{\rm{lr}} - {\rm{a}}}}{{{\rm{r}} - 1}},{\rm{r}} \ne 1$

Example 1

Find the 3 arithmetic mean between 7 and 23

a.

Here

First term (a) = 7

Last term (b) = 23

No of means = 3

Common difference (d) =$\frac{{{\rm{b}} - {\rm{a\: }}}}{{{\rm{n}} + 1}}$

=$\frac{{{\rm{\: }}23 - 7{\rm{\: }}}}{{3 + 1}}$ = 4

Now, m1=a+d =11

m2=a + 2d = 15

m3=a+3d =19

Example 2

If the third and the eleventh terms of an arithmetic sequence are 8 and -8 respectively, Find the first seven terms of sequence.

Soln

Given,

t3=8

t11=-8

we have ,

tn=a + (n-1 )d

Now,

t3= a + (3-1 )d

or , 8=a + 2d………………..1

t11= a + ( 11 -1 )d

-8 = a+ 10d………………2

Solving 1 and 2

We get a = 12 and d = -2

t1=12

t2 =12 + (2-1) *- 2= 12 -2= 10

t3 =12 + (3-1) *-2 = 12 -4 = 8

t4=12 + (4-1 )* -2 = 12 -6 = 6

t5= 12 + (5-1)*-2 = 4

t6= 12 + (6- 1 )*-2= 12-10 = 2

t7= 12 + (7- 1) *-2 = 0

Relation and functions

A relation is any set of ordered pair (x,y)such that the valueof the second coordinate y’ depends on the value of the first coordinate x’ then y is the dependent variable and x is the independent variable.

Different ways of representing relationship

a. Mapping diagram

b. Set of ordered pairs

c. Description

d. Table

e. Graph

Function

A function is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2.

Composite function

Let f : A → B and g: B → C be the two functions .the function gof: A→ C is called composite functionfrom A to C

Example 3

Iff = {(1,2),(2,3) (3,4) } anf g ={ (2,a)(4,c) ,(3,b)} , then show that the composite function gof in arrow

diagram and find the it in ordered pair form.

Soln

gof={(1, a) ,(2,b) , (3,c)}

Inverse function

If a function f(x ) =yis injective, exactly one functionwill exist such that  , otherwise no such function will exist .The functionis called the inverse function ofbecause it "reverses"  ; that is to say  .

Types of function

Onto function

A function f from a set X to a set Y is onto, if every element in Y has a corresponding element in X such that f(x) = y.

Into function

A function f from a set X to a set Y is into, if element in Y is proper subset of X

One to one on to function

A function f from a set X to a set Y is One to one on to function, if every element in Y uniquely assign to element of X.

One to one into function

A function f from a set X to a set Y is One to one into function, if element in Y not necessarily assign to element of X.

Many to one on to function

A function f from a set X to a set Y is Many to one on to function, if element in Y has more than one element in x mapped to them.

Many to one into function

A function f from a set X to a set Y is Many to one into function, if element in Y has at least one element which is not mapped to the element of X.

Domain and Range

The domain is the set of all first elements of ordered pairs (x-coordinates)

The range is the set of all second elements of ordered pairs (y-coordinates).

Rational and irrational numbers

A rational number is the set of fractional numbers. It is denoted by Q. we can define

Q = {x: x =${\rm{\: }}\frac{{\rm{P}}}{{\rm{q}}}$ p,qϵ Z and q ≠ O}

An irrational number cannot be written as a fraction. An irrational number has endless non-repeating digits to the right of the decimal point.

A surd is an irrational number, a number which cannot be expressed as a fraction or as a terminating or recurring decimal. It is left as a square root. It can be written as${\rm{\: }}\sqrt[{\rm{n}}]{{\rm{a}}}$

Laws of radicals:

${\left( {\sqrt[{\rm{n}}]{{\rm{a}}}} \right)^{\rm{n}}} = {\rm{a}}$

$\sqrt[{\rm{n}}]{{\rm{a}}}$ × $\sqrt[{\rm{n}}]{{\rm{a}}}$ =$\sqrt[{\rm{n}}]{{\rm{a}}}$b

$\frac{{\sqrt[{\rm{n}}]{{\rm{a}}}}}{{\sqrt[{\rm{n}}]{{\rm{b}}}{\rm{\: }}}}$=$\sqrt[{\rm{n}}]{{\frac{{\rm{a}}}{{\rm{b}}}}}$

$^{\rm{m}}\sqrt {\sqrt[{{\rm{\: n}}}]{{\rm{a}}}} {\rm{\: }}$=$\sqrt[{{\rm{mn}}}]{{\rm{a}}}$

Surds having the same order and radicals can be added or subtracted.

Eg5√ 11 + 2√ 11 =7 √ 11

Rationalization

When the surd is multiplied by its conjugate we get the rational numbers

Eg – $\frac{1}{{1 - \sqrt 3 }}$

$= {\rm{\: }}\frac{1}{{1 - \sqrt 3 }}* \frac{{1 + \sqrt 3 }}{{1 + {\rm{\: }}\sqrt 3 }}{\rm{\: }}$

= $\frac{{1 + \sqrt 3 }}{{1 - 3}}$

= - $\frac{1}{2}$- $\frac{{\sqrt 3 }}{2}$

Polynomial:

A polynomial is an expression consisting of variables and coefficients which only employs the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single variable x is x2 −x + 3. An example in three variables is x3 + 2xy2z2 − yz + 1.

Degree of polynomial: The degree of the polynomialof one variable is the value of the largest exponent of the variable.

The Remainder theorem

If P(x) of degree n ≥ 1 divided by x-r , the remainder is P (r).

Consider P(x) = (x − r) q(x) + R

Note that if we let x = r, the expression becomes

P(r) = (r − r) q(r) + R

Simplifying gives:

P(r) = R

This leads us to the Remainder Theorem which states:

If a polynomial P(x) is divided by (x − r) and a remainder R is obtained, then P(r) = R.

Example 4

${{\rm{x}}^3} + {\rm{\: p}}{{\rm{x}}^2} + {\rm{qx}} + 5{\rm{\: }}$Leaves remainder 1 when divided by x+ 2 and 16 when divided by x -1 , find p and q

Soln

letP(x) = ${{\rm{x}}^3} + {\rm{\: p}}{{\rm{x}}^2} + {\rm{qx}} + 5{\rm{\: }}$, R = 1 and Q(x) = x+2

P(-2) =-8 + 2p -2q +5

1 = 2p -2q -3

4 = 2p -2q

2 = p- q ……………………………………….1

P(x) = ${{\rm{x}}^3} + {\rm{\: p}}{{\rm{x}}^2} + {\rm{qx}} + 5{\rm{\: }}$, R = 16 and Q(x) = x-1

P(1) = 1 + p+q+5

16 = 6+p +q

10= p + q ……………………2

Solving 1 and 2 we getp = 6 and q =4

The Factor theorem

If P(x) of degree n ≥ 1 has remainder P(r) = 0, then (x-r) ids the factors of P(x)

The factor theorem is a theorem linking factors and zeros of a polynomial. It is commonly applied to factorizing and finding the roots of polynomial equations. The theorem states that is a factor of a polynomial P(x) if; that is, r is a root of P(x).

Example5

Let P(x) = x3 –kx2 –x - 6 = 0

D(x) = (x -2)

If the D(x) = (x -2) is the factor of the given polynomial the remainder is Zero

By synthetic division

As the remainder is zero (x-2) divides P(x) perfectly and therefore it is one of the factors of P(x)

i.e4K+4= 0

K = -1

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