ELEMENTARY MATH ASSIGNMENT

7/30/2019 ELEMENTARY MATH ASSIGNMENT

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TABLE OF CONTENTS:TABLE OF CONTENTS:TABLE OF CONTENTS:TABLE OF CONTENTS:

SERIAL SUBJECT CO-SUBJECT PAGE

1. NUMBRERSYSTEM 3-4

WHATISDIGIT? 3

WHATISINTIGERS? 3

WHATISRATIONALANDIRRATIONALNUMBER? 3

WHATISPRIMEANDCOMPOSITENUMBER? 3

WHATISREALNUMBER? 3

2. COMPLEXNUMBER 4-6

WHATISCOMPLEXNUMBER? 4

PROPERTIESOFCOMPLEXNUMBER. 4

PROVETHAT2ISIRRATIONAL. 5ABSLUTEVALUE 6

3. INEQUALITIES 6-7

INEQUALITIES 6

INTERVAL 6

EQUATIONANDIDENTITYANDTHEIRDIFFERENCE 7

4. LINEAREQUATION 7-9WHATISLINEAREQUATION? 8

EXERSIZES 8

5. QUADRATICEQUATION 9-14

QUADRATICEQUATION 9

QUADRATICFORMULA 9

DERIVATIONOFQUADRATICFORMULA 11

THE NATURE OF THE SOLUTION OF A QUADRATIC EQUATION 12

6. FUNCTION 14-17

EXERSIZES 14

EVEN FUNCTION 17ODDFUNCTION 17

7. INDICESAND LOGARITHEM 17-24

8. NATUREOF SOLUTION LINEAR

EQUATION:24-28

9. GUSSIANLAW 29-31

10. SERIES 32-47

ARITHMATICPROGRESSION 32

GEOMATRICPROGRESSION 32

11. EXTRAQUESTIONS 48-50


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NUMBER SYSTEM:NUMBER SYSTEM:NUMBER SYSTEM:NUMBER SYSTEM:

Digit:

Digits are the integers 0 through 9 of which other number are comprised.

Integers:

The set of integer consists of natural number or positive integers, zero and the negative integers.

The set of integers is denoted by Z and defined as, Z={-.-2,-1,0,1,2,3}

*[set is expressed through second bracket]

Rational number:

A rational number is a number can be put in the form , where p and q are integers and q is not equal

zero.

The set of rational number is denoted by Q and defined as Q= { , p, q Z and q0}

Irrational number:

A rational number is a number can be put in the form

, where p and q are integers and q is not equal

zero.

The number which is not rational is called irrational number. The set of Irrational number is defined Q

For example: 2, 3, 5, etc.

Prime number:

A number greater than 1 which is not exactly divisible by any numbers except itself and unity is called a

Prime number.

For example: 2, 5, 13, 23, etc.

Composite number:

A number which is divisible by other numbers besides itself and unity is called a composite number.

Real number:

The collection of all the rational and irrational numbers is called the system of real numbers.

It is denoted by R and so R= {Q U Q }

There are three kinds of rational fraction:


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- If the upper digit is smaller than lower digit than proper fraction. - If the upper digit is bigger than lower digit than improper fraction. 1 - than it will be Mix proper fraction.

Complex number:

Complex number:

The set of all complex number is denoted by C and is defined by C= {a+ib: a, bR and =-1}, where a iscalled the real part and b is called the imaginary part of complex number and is the imaginary unit.

Example:5+i6,3-4i,9+11i etc.

Properties of complex number:

The addition and subtraction of complex is again a complex number.

The multiplication and division of complex number is again a complex number.*[only the term of conjugate complex multiplication of complex number is not a complex number]

REAL NUMBER

RATIONAL

NUMBER

IRRATIONAL

NUMBER

INTEGER RATIONAL

FRACTION

PROPER IMPROPER MIXED

POSITIVE NEGATIVE ZERO


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EVALUATE

in form.

Solution,

= = = .=

= = =

*Prove that 2 is irraonal.

Proof:

Suppose, that 2 is a raonal number.

Then, 2=, where p and q are integers and q 0

Again, suppose that the rational number is in its lowestb term.

*[lowest term: it is not possible to divide]

i.e.: p and q have no common factors, without 1.squaring both side of the above equation we get,

2=

Or, =2 The term 2 represents an even integer, so is an even integer, and hence p is an even

integer, say p=2r, where r is also an integer.

Replacing, p by 2r in the equation =2 we get,2 =2 Or, 4 =2 Or, 2 =


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The term 2 repreesnts an even integer .so is an even integer and hence q is an even integer, Thuswe have seen that both p and q are even integers. That is they have a common factor 2, which,

contradicts our assumption that p and q have no common factors. Hence it, follows that 2 is not a

rational number. Therefore, 2 is an irraonal number.

[Proved]

Absolute value:

The absolute value of a real number a is denoted by lal

and defined as,

a =

, 0

, 00, 0

Inequalities:

Inequalities:

A number a is greater than another numberb if (a-b) is positive which denoted by a>b

Again a is smaller than b when(a-b) is negative and we denote it by a


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[a,b] axb 0 A b

0 a b

Interval inequality notation graph

[A,b) ax


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Solve the linear equalities and draw the graph.

Solution,

: : : }

Graph:

-2 13

*same number addition +subtraction=order no change

*same negative number multiplication+division=order change

:Quadratic equation:

Quadratic equation:

A quadratic equation in x is an equation that can be written in the standard form, 0where a,b and c are real numbers with a0

Example: 2 5 9 05 3 0


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Quadratic formula:

The solution of

0with (a0) are given by the quadrac formula,

42 [*x ]

Solve,

3 9 0Solution,

3 9 0. 1

, 1 0, , 1 3 9

We know,

42

3 3 4.1.92.1 3 9362 3 452

3

5 9

2

3352

Either, or, 3352 3352

Solve,


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2 5 0Solution,

2 5 0 . 1, 1 0, , 2 1 5

We know,

42 1 1 4.5.22.2

1 1404 1 414

Either, or, 1 414 1 414

Derivation of quadratic formula:

We have,

0 . . .

2. .


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2. .

Which is the required quadratic formula.

There are three kinds of root:

1. Real-the root where number is available.2. Imaginary-where i is available3. Repeated root-x =,

The nature of the solution of a quadratic equation:

The nature of the solution of quadratic equation, 0, 0can bbe classified by thediscriminant -4ac as follows:

1. If -4ac>0, the equation has two distinct real solution.2. If -4ac


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.

6 11 15 06 11 15 0 . 1, 1 0, ,

6 11 15Discriminant of (1) is,

= 11 4.15.6= 121360 = 239= 239 0 4 239 0

.We know,

42 = ... = =

= =

Describe the nature of the roots of the equation and then solve it.

13 7 1 013 7 1 0 . 1


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, 1

0, ,

13 7 1

Discriminant of (1) is,

= 7 4.13.1= 4952 = 3=

3 0

4 3 0 .We know,

42 = . = = .

Independent variable:

3x+9y=1

Or, 9y=1-3x

Or, y=

Here, x is independent and y is independent variable.


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FUNCTION:

Function:

A function is a special type of relationship between a dependent and independent variable.

For example: father and son, office and employ

Domain and range:

13x+5y+7=0

Or, 5y=7-13x

Or, y= Independent variable is domain so x is domain

Dependent variable is range so y is range.

Explicit function:

If a function expressed directly in terms of independent variable is called explicit function.

Example: 3x+9y+7=0, x+y=5 etc.

Implicit function:

If a function cannot expressed directly in terms of independent variable is called implicit function.

Example:

,log

If

5 9

Let, , . 1


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,

5 9

5 9 1 ,

5 9 5 9 95

/ 95

[Solve]If

/

Let, , . 1 ,

1319

/

1723

1319 / 1723 1 ,

/ / /


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/

/ 437 323299

Even function:Even function:Even function:Even function:A function given by x is even if for each x the domain such that x -xExample:

I. The function, (x) is evenII. The function, (x) is even

ODD function:ODD function:ODD function:ODD function:

A function given by (x) is odd if for each x the domain such that (-x) -(x)Example:

I. The function, (x) is odd.II. The function, (x) sin is odd.

Exponential FunctionThe exponential function f with base a denoted by f(x) = a

x; where a>0, a 0 and x is only real

number.

Example: 2x, 5x-1 etc

Natural exponential function

The function f(x) = ex

is called the natural exponential function where x is variable.

Example: e5x

, e3x+7

etc


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Indices and Logarithm:

Problem:ex

= 7

Solution: ex

= 7

Taking ln both side we get

ln(ex)

= ln(7)

or, xlne = ln (7)

or, x * 1= ln 7

or, x= ln 7

NOTE:

lnx = p

or, x = e

p

lnx = -3

or x = e-3

Problem: 5+2lnx = 4

Solution: 5+2lnx =4

2lnx = 4-5 2lnx = -1 lnx = - x = e x = x =

Solve


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Problem: e2x

- 3ex

+ 2 = 0

Solution: e2x

- 3ex+ 2 = 0

(ex)2 - 3ex + 2 = 0Let e

x= p

p2 - 3p + 2 = 0 p2 - 2p p + 2 = 0 p(p-1) - 1(p-1) = 0 (p-2) (p-1) = 0

Either, or,

p-2= 0 p-1= 0

p=2 p= 1 ex = 2 ex= 1 x = ln2 x= ln1 x = ln2 x= 0

The required solution: x=0, ln2Logarithmic Function

Logarithmic function:- for x 0 and a 1 the function given by f(x) = logax is called thelogarithmic function with base a.

Example: f(x) = log2x

is a logarithmic function with base 2


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Properties of Logarithms

1. loga1 = 02. logaa = 13. loga = xlogaa = x4. logax = logayx = y5. logax = b x = ab6. log10(MN) = log10M + log10N7.

log10(M/N)

= log10M

log10N

Problem: 23X-2

= 32X+1

Solution: Taking logarithm on both sides we get

Log2(3x-2)

= log3(2x+1)

(3x-2)log2 = (2x+1)log3

3xlog2 2log2 = 2xlog3 + log3

3xlog2 - 2xlog3 = log3 + 2log2 x(3log2 2log3) = log3 + 2log2

log3 + 2log2

x = 3log2 2log3log3 + log2

2

=

log23 log3

2

log3 + log4=

log8 + log9


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log(34)=

Log(8/9)

log12 x =log(8/9)

This is the required solution.

Problem: Find the value of

/.

.

.

. . /3/. ( )1/5 . ( )1/7

= .

( )1/8. ( . )1/5 /

/. /. / = .

/. y .

/

/

. = . y

= . y/

/= . y//

= . y/= . y


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= 1 . y

= y

This required solution.

Problem: If log = (log + log y) then show that + = 23.Here,

Log = (log + logy) log

=

log

1/2

= 1/2 = 2 =xy

=xy +2xy+y2 = 25xy x2 + y2 = 23xy

+

=

+ = 23[SOLVED]

Problem: If . . = then show that, a b = 1.Here,

. . = ... = ..

. =


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...

=

.. = .. = . = 27 . 1 = 27 = 27 b-a = -1 [ Multiplying both side with -1] a-b = 1

[Showed]

Problem: If log = ( log a + log b) then shoe that, a = b

Here,

log = ( log a + log b)

log

=

log ab

log = log (ab)1/2 = = ab [ Squaring both side] = ab


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2 = 4ab 2 4 = 0 2 = 0 (a-b = 0 a-b = 0

a = b

[Showed]

Find the value of ,log2 [log2 { log3 (log3)}]

= log2 [log2 { log3 ( }]= log2 [log2 { log3 (

3)}]

= log2 [ log2 (log39)}]

= log2 [log2 ( log33)}]= log2 [log2

2]

= log21

= 0

If = 0 and x+y 0 then shoe that log ( x+y) = (logx + logy + log3)Here, = 0

(x+y 3xy (x+y) = 0 (x+y) { (x+y 3xy}


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NATUREOF SOLUTION LINEAR EQUATION:

But, x+y 0

(x+y

= 3xy

(x+y) = (3xy log (x+y) = log (3xy) ( x+y) = (logx + logy + log3)

[Showed]

System of equation:

ax + by + cz R1

a1x+b1y+c1z R2

a2x+b2y+c2z R3

c1 c2 c3ORDER SYSTEM OF LINEAR EQUATION

The system of linear equation of 33 is given by:

Solution:

I. Unique solutionII. Infinite number of solution

III. No solution

Number of equation = number of variable = Unique solution4x 9y = 7 (i)

7x + 5y = 6(ii)


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Number of equation number of variable= Unique number of solution.3x + 7y +9 = 0

9x + 4y + 6 = 0

No solution:Problem: -x-2y = 3 ..(i) 2

-2x+4y = 1 .(ii) 1x-2y = 3

x= 3+2yPutting the value of x in the equation (ii) we get

-2 (3+2y) + 4y = 1

-6 - 4y + 4y = 1 0 = 7So, the equation has no solution.

Describe the nature of solution of a system of liner equation.2x+3y=1

3x-y = 4

Solve by:

i. Method of substitution.ii. Method of elimination.

iii. Method of cross multiplication.

Method of Substitution:2x + 3y = 1 .(i)

3x y = 4 (ii)

From the equation (ii), we get,


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3x y = 4

- y = 4 3x Y = 3x 4 ..(iii)Putting the value of (iii) in equation (i), we get,

2x + 3 (3x 4) = 1

2x + 9x 12 = 1 11x = 13 x =

y = 3 . 4= 4

=

=

Method of Elimination:2x + 3y = 1 .(i)

3x y = 4 .(ii)

Multiplying (i) 1 and (ii) 3 and adding them, we get,2x +3y = 1

9x 3y = 12

11x = 13 x =

Putting the value of x in equation (ii) we get,

3 . y = 4

y = 4


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y = -

Method of cross Multiplication

x + 3y 4 = 0 ..(i)

3x y 3 = 0 ..(ii)

By the law of cross multiplication, we can write

= = = = x = = y =

The required solution: x = , y =

Problem: 3x 10y +5 = 0 ..(i)

2x 7y + 20 = 0 (ii)

Method of Substitution:

From the equation (i) we get

3x -10y + 5 = 0

3x = 10y -5 x = ..(iii)

Putting the value of (iii) in (ii), we get,

2 .

7y + 20 = 0


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= 0 20y 10 -21y +60 = 0 - y 10 = -60 y + 10 = 60 y = 50

x =.

=

= 165

The solution of the equation is ( x,y ) (165,50)

Method of Elimination:

3x 10y +5 = 0 .(i)

2x 7y + 20 = 0 .(ii)

Multiplying (ii) by 3 we get

6x 20y +10 = 0

6x 21y +60 = 0

(-) (+) (-)

Y 50 = 0

y = 50Putting the value of y in equation (i) we get,3x 10 . 50 + 5 = 0

3x = 500 5 x = x = 165


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GUSSIAN LAW:

The solution of the equation is ( x,y ) (165,50)

Method of cross Multiplication:

3x 10y +5 = 0 .(i)

2x 7y + 20 = 0 .(ii)

By the law of cross multiplication we get,

= = = =

Either, = x =

x = 165or,

- =- y = 50

The required solution is (x,y) : (165,50)

3x + 4y +5z = 7 R19x + 5y + 4z = 5 R2

7x + 8y + z = 16 R3

c1 c2 c3

Solve the following linear equation:


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2x + 4y + z = - 4

2x 4y + 6z = 13

4x 2y + z = 6

2x + 4y + z = - 40 - 8y + 5z = 17 [R2 = R2 R1]

4x - 2y + z = 6

2x +4y +z = - 40 8y + 5z = 17

0 10y z = 14 [R3 = R3 2R1]

2x +4y +z = - 4 .(i)0 8y + 5z = 17 .(ii)

0 0 58z = -58 ..(iii) [R3 = 8R3 10 R2]

From the equation (iii), we get

-58z = -58

z = z = 1

Putting the value of z in equation (ii) , we get

-8y + 5 .1 = 17

-8y = 17 5

-8y = 12 y = -Putting the value of y and z in equation (i) , we get

2x + 4 + 1 = -4 2x 6 + 1 = -4 2x 5 = -4


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x = The required solution is (x,y,z) : (, -,1) - x 2y + 3z = 9

-x 3y + 0 = - 4

2x 5y + 5z = 17

- x 2y + 3z = 90 + 5y 3z = -13 [R2 = R2 R1]

2x 5y + 5z = 17

- x 2y + 3z = 90 + 5y 3z = -13

0 9y + 8z = 26 [R3 = R3 + 2R1]

- x 2y + 3z = 9 .(I)0 + 5y 3z = -13 .(II)

0 0 + 13z = 13(III) [R3 = 5R3 + 9 R2]

From the equation (iii) we get,

13z = 13 z = 1Putting the value of z in equation (ii) we get,

5y 3 .1 = - 13

5y = - 13 + 3 y = -2

Putting the value of z , y in equation (i) we get,

- x 2 (-2) + 3 . 1 = 9 - x + 4 + 3 = 9


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- x = 2

x = - 2

The required solution is (x , y , z) = (-2 , -2 , 1)

SERIES

Arithmatic progression:( )::::Arithmatic progression is a progression which is increasing ordecreasing by a common difference.

Example:

1+3+5+7+

d=3-1=2

Geometric progression( ) :geometric progression is a progression which is increasing or by a

common ratio.

Example:

1+3+9+27+

Arithmetic progression:

GT a + (n-1) d Common defense

a n

Fast term number of term

Sum = {2a + (n-1) d} 1 + 2 + 3 + ..+ n =

12

+ 2

2

+ 3

2

+ .+ n

2

=

13 + 23 + 33 + .+ n3 = { 12 + 32 + 52 + (2n -1)2 = n2


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Geometric progression( ) :

GT = aqn-1 Number of term

1st term Common ratio

Sum = . q 1

= , q 1

1 + 3 + 9 + 27 + ..

Sum = , r 1=

, r1a = 1st term

r = common ratio

Find the value of the series

1 , - , -

=1 ( , 2

- 3

..

= + 2 , 3r = -

1a = 1

1st term a-1

Common ratio = = 1r 1 we knowSum =

=

=


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=

= Ans.

Problem:8 , 42 , 4 to infinity1st term a = 8

Common ratio r =

=

.

= 1 = r 1

we know ,

Sum ==

=

=

=

[ Multiplying denominator and nominator by 21 ]= 16 + 82= 27.3137 Ans.

Problem: Find the sum of the series 72 , 70, 68 40Here,

First term, a = 72

Common difference = 70 72

= -2

Let,

nth term be 40


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We know,

nth term = a + (n 1) d

40 = 72 + (n 1) (-2) 40 = 72 2n +2 40 = 74 2n 2n = 74 40 2n = 34

n = 17

Sum = { 2a + (n -1) d }= { 2. 72 + (17 1) (-2)}

= { 144 + 16 (-2)}

= (144 32)

= 112

= 952

Problem: 21 + 15 + 9 + .+ (-93)Here,

Frist term a = 21

Common difference d = 15 21

= - 6

Let,nth term = a + (n-1) d

-93 = 21 + (n -1) (-6) -93 = 21 6n + 6 - 93 = 27 6n 6n = 27 +93 6n = 120


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n=

n = 20 Sum = { 2a + (n -1) d }= { 2. 21 + (20 1) (-6)}

= 10 { 42 + 19 (-6)}

= 10 (42 144)

= 10 (-72)= - 720 Ans.

Problem: 7 + 11 + 15 + .. sum up to 15 term the first term of a series in A.P is 17 ,the last term 12

and the sum 25 , find the number of term and common difference.Let,

The number of terms be n

And common difference be d

Here,First term a = 17

nth term = - 12

Sum = 25

We know,

nth term = a + (n-1) d

- 12

= 17 + (n-1) d

- = 17 + (n-1) d (n-1) d = - 17 (n-1) d =

(n-1) d =- ..(i)Again,


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Sum = { 2a + (n -1) d }

25

=

{ 2

17 + (n-1) d}

= { 34 + (n-1) d} = 34 [using equ .1] = = = 407 = 37n 37n = 407 n =

n = 11Putting the value of n in equation (i)

(n-1) d =

(11-1) d =

10 d = d =

d = Number of system = 11And common difference =


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Problem: 7 +11 + 15 + sum up to 15 term.Here,

First term a = 7

Common difference d = 11 7

= 4

We know,

Sum = { 2a + (n -1) d }

= { 2 7 + (15 -1) 4 }

= (14 + 14 4)= (14 + 56)

=

= 525Ans.

G.P

GT aqn-1

Sum: : q 1 : q 1

Infinite: sum ,

,

Problem: The third term of a G. P and 6th term is find the 8th term.Solution,


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Let the 1st

term of the G.P be a

And common ratio of the G.P be q

Third term = aq3 1 = aq2A/Q aq2 =

..(i)Again,

6th term = aq6 1

A/Q, aq5 = (ii)

Dividing equation (ii) by equal (i)

we get,

=

q3 = q3 =

q3 = 3 q =

Putting the value of q in equation (i) we get

a 2 = a

=

a = 9 a = 6 8th term = aq8 1

=aq7

= 6

7


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= 6

= Ans.

Problem: The 4th term of a G.P is and 7th term is - find the 6th term.Solution,

Let the 1st term of the G.P be a

And common ration of the G.P be q

4th

term = aq4 1

= aq3

A/Q, aq3 = (i)

7th term is = aq7 1

= aq6

A/Q, aq6 = -(ii)

Dividing equation (ii) by equal (i)

=

q3 = - 81 q3 = - q3 = - q = 3 q = -

Putting the value of q in equation (i) we get,

aq3 =


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a 3 = - = a = - - 27 a = -

6th

term aq6 1

= aq5

= - 5= -

- =

Ans. Problem: Find the sum up to n term of the series: 11 + 102 + 1003 + 10004 + .

Let,

Sum of the nth term be Sn

Sn = 11 + 102 + 1003 + 10004 + .to n term= 10 + 1 + 100 +2 +1000 + 3 10000 + 4 + to n term

= 10 + 1 + 102 + 2 + 103 + 3 + 104 + 4 + to n term

= ( 10 + 102 + 103 + ..... to n term) ( 1+ 2 +3 + 4 + to n term)

q 1

1 1

= +

= ( 10 1 + Ans.


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Find term up to n term of the series: 8 + 88 + 888 + 8888 + ..Let,

Sn be the sum of the nth term

Sn = 8 + 88 + 888 + 8888 + .n term 8(1 + 11 + 111 + 1111 + to n term) (9 + 99 + 999 + 9999 + . to n term) 10 1 + 100 1 + 1000 1 + 10000 1 + to n term

10 1 + 102

1 + 103

1 + 104

1 + .. to n term

{ ( 10 + 102 + 103 + 104 + .+ 10n) (1 + 1 + 1 + 1 + . to n term)}

= Sn = (10n 1) -

8 + 88 + 888 + 8888 + .to n term

(10n 1) -

Find the 12th term of 2 , -2 , 6Here,

a = 2

q = = - 3

12th term = aq12 1

= 2 ( - 3)11= 2 - (420 8883)


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= -841.776692 Ans...

Find the sum up to n term of the series 0.5 +0.55+ 0.555+ Let

Sn be the sum of the series to n term

Sn = 0.5 + 0.55 + 0.555 + to n term= 5(0.1 + 0.11 + 0.111 + to n teem)

= (0.9 +0.99 + 0.999 + + to n term)= ( 1 - .10 + 1 - .01 + 1 - .001 + .. to n term)

= ( 1 + 1 + 1 + to n term ) ( 0.10 + 0.01 + 0.001 )

= { n - }

=

Sn =

= 1

= 1

=

-

1

The third term of a g.p isand 6th term is ,find the 8th term.

Solution,

Let, the 1st term of the g.p be = a

And common ratio of the g.p be=q


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3rd term= =

According to question,

23Again,

6th term= = According to question,

281Dividing equation (2) by equation (1), we get,

28123

281 23 13 13

Putting the value of q in equation (1), we get,

. 13

23

. 19 23

23 91

6 8th term


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= = = 6. = 6. =

Find the sum up n term of the series:

11+102+1003+10004+..

Let, sum of the term of the series be .

11 102 1003 10004 ..

10 1 100 2 1000 3 10000 4 .. 10 1 10 210 310 4 . .

10 10 10 10 ..tonterm + (1+2+3+4..tonterm 1010 1101 12

109 10 1 12 Sum of the term of the series:0.777+0.77+0.7+

Let, sum of the term of the series be . 0.777 0.77 0.7

70.111 0.11 0.1 .

79 0.999 0.99 0.9 . .

79 1 0.001 1 .01 1 .1 . . 1 1 1 1 . . 0 .0010.010.1.


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47

79 11000 10 1

101

79 11000 10 19

79 79 19000 10 1 79 781000 10 1

A firm pays taka 4000 to its manager, the manager is given an increment of tk. 500 every year.

(1) Find the total salary paid to the manager in 10 years.(2) Salary of the end of 10 years.

Here,

1st term, a =4000

Number of term, n=10

General termed=500

Total salary in ten years:

2 2 1 102 2.4000101500 102 80004500 5 12500 62500Salary of the end of ten years: 1 4000 101500

8500

A person has two parents, four grandparents, eight great grandparents etc.

Find the number of ancestors which a person has in the 12th generation back and total number of all

ancestors in those proceeding.

Here, number of parents of 1st, 2nd and 3rd generation are 2, 4, and 8 respectively.

1st term = 2

Common ratio,q = 2 1


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12th

generation,

= = 2 2= 2= 4096

Total number of ancestors in their proceeding to 12th generation,

1 1 22 12 1 2409612 1 2 4095 8190

EXTRA QUESTIONS:

1. The term of p,p+d,p+2d,p+3d,..Here,

a=p

d= p+d p

d=d.

We know,

nth term = a + (n-1) d

p+ 1d

2. If ,thanc . .

1 1


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1 1 3. What is the of 3+9+27+.to n term = ?

We know,term = Here,

a=3

q= 3 1term=3.3= 3= 34. If ?

Given,

Ans.5. How many neutral number lies between 25 to 59.

There are 8 neutral number lies between 25 to 59 they are:

29,31,37,41,43,47,53,59

6. How many root in a quadratic equation?There are 2 root in a quadratic equation they are:

7. How many even prime in real number set?There is one even prime in real number set and that is 2.

8. When the roots of Q.E become real and equal?When 4 =0 than the roots of Q.E become real and equal.


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9. .

.

We know,

12 16 Here,n=20

202012.201

6

2019396 2470

10.General term of A.P and G.P.General term of A.P:

1 General term of G.P:

THANK YOU

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ELEMENTARY MATH ASSIGNMENT