QUIZ OF THE DAY

Find the next two terms of the sequence :

2, 4, 8, 16, …..                      

Arithmetic Progression

Arithmetic Progression

Topic -1 Sequence & Its Types

Quick Review –

Ø  Sequence : A Sequence is a collection of numbers arranged in a definite order according to some definite rule.

Example :

(i)  1, 4, 9, 16, ……(Collection of Perfect squares of Natural numbers)

(ii) 1, 3, 5, 7, …….(Collection of Positive odd Integers)

(iii) −2, − 4, − 6, …..(Collection of Negative even Integers)

Ø  Types of Sequence :

(a) Finite Sequence :

If the number of terms in a sequence is finite (Countable), i.e.,

 if there is an end of terms in the

     sequence, then it is called a finite sequence.

e.g., : 1, 2, 3, ….. 20.

(b)  Infinite Sequence : 

If the number of terms in a sequence is infinite, then it is called an infinite sequence.

      

 e.g., : 1, 3, 5, 7, ………..

Pythagoras Theorem

Pythagoras Theorem

  

A ladder reaches a window which is 

15 metres above 

the ground on one side of the street. 

Keeping its foot at 

the same point, the ladder is turned to the other side 

of 

the street to reach a window 8 metre high. 

Find the width of the street, if the length of the 

ladder is 17 metre.

QUIZ OF THE DAY

QUIZ OF THE DAY

Find the next two terms in the sequence :

1, 3, 7, 15, 31, …..       (March 2013)

Equations in One Variable

study the lesson:

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 comments and answers below:

Each one of the following is an equation is one variable. There are terms with variables on both sides of the equation. The equations have been solved using the rules of equality. Study them.

              Method                                                     Explanation

Ex. 1.  8y + 5 = 3y + 20              Simplified form using the             

∴ 8y – 3y + 5 = 3y – 3y + 20      subtraction property of an

∴ 5y + 5 = 20                                 equation.

∴ 5y + 5 – 5 = 20 – 5

∴ 5y = 15                                       Simplified form using the division

∴  =    ∴ y = 3                property of an equation.

                        Solution of the equation : 3

Solve the equations.

(1)  y – 2 = 9

(2)  p + 3 = 12

(3)  3x = 18

(4)  m/4 = 8

The Use of letters in place of Numbers

The Use of letters in place of Numbers

        We use symbols in mathematics. Using symbols, we write ‘the sum of five and nine’ in short as ‘5 + 9’.

         

Using symbols makes our writing short and clear. Use of letters, too makes mathematical writing easier.

         

Letters are used in place of numbers in two ways.

1.    The use of a letter for an unknown number.

‘ what is the number that is bigger than 8 by 4 ?” To find the answer to this question, let us add ‘8 + 4’. The sum 12 is the answer to the question.

      

 Using a sign, we write the information number that is bigger than four by eight’ as ‘8 + 4’.

      

 Now, let us see how we could write the information ‘a number that is bigger than another number by 4’ in signs.

      

We do not know exactly which number we are speaking of still, we can write this information using signs. Let us show the unknown number with the letter ‘a’. using signs, we can write ‘a’ number bigger than another number by 4’ as ‘a

+ 4’.

        

Similarly, ‘a number less than another number by 7’ can be written as ‘b – 7’. Here, b is written in place of the unknown number.

        

You will notice that when we add, subtract, multiply or divide two or more numbers, we get only one number in the end. But, the expression we get by using the sign for an operation between a letter and a number cannot be simplified any further.

     

   ‘The product  of a number and 2’ can be written as ‘m × 2’ or          2 × m by writing ‘m’ in place of ‘a number’. However, the product of a number and a letter is conventionally written with the number first and without the multiplication sign. Thus, the product m × 2 or 2 × m is written as 2m. similarly, 10 × k or k × 10 is written as 10k.

         

Some more examples of the use of a letter in place of an unknown number are given below. Study them. Note that we can choose to write any of the letters a,b,c,…,z in place of numbers.

   Information                                                      Expressed using letters

(1)  Sum of 10 and another number                              10 + p

(2)  Product of 23 and another number                         23 × d or 23d

(3) The quotient of 18 divided by a number              18 ÷ y or 18/y

(4)  The quotient of a number divided by 18           x/18 or x ÷ 18

(5)  The number obtained by subtracting                  a - 15           

15 from a number

(6)  The number obtained by subtracting                      15 – b          

a number from 15

(7)  A number less than another by 2                             (d – 2)

(8) The mangoes left after eating 6 from a                    (m – 6)               

Certain number of mangoes       

 

(9)  The number of guavas in each share if                           

3 equal shares are made of a certain                        p/3             

Number of guavas.

Order of Operations and the Use of Brackets

Order of Operations and

 the Use of Brackets

You have learned to carry out the operations of addition, subtraction, multiplication, and division. Let us study these operations further. Consider the following examples :

Ex (1) Simplify : 75 + 25 x 10

Addition and multiplication are the two operations in this example.

If we carry out the addition 75 + 25 first, the value of this expression will be 100 x 10 = 1000

If we carry out the multiplication 25 x 10 first, then the value of the expression w

ill be 75 + 250 = 325.

We shall have to say that the same expression has two different values. It is obvious that this will cause confusion.

Use of Brackets

It is necessary to know which operation should be carried out first in order that the expression yields a single value. Brackets are used for this purpose. The operation to be carried out first is written inside brackets.

             

For example, if the addition is to be carried out first in the expression 75 + 25 x 10, then the expression is written as                   (75 + 25) x 10.

 If the expression is 75 +(25x10), multiplication will be carried out first.

The value of the expression (75 + 25 ) x 10 is 1000.

And the value of 75 + (25 x 10 ) is 325.

Similarly, ( 48 ÷ 8 ) ÷ 2 = 3 and 48 ÷ ( 8 ÷ 2 ) = 12.

The following rules are generally

 followed to minimize the use of 

brackets in an expression.

(1)  Operations in brackets should be carried out first.

(2)  After that, multiplication and division must be carried out in the order in which they occur from left to right.

(3)  Then, addition and subtraction should be carried out in the order in which they occur from left to right.

(4) If there are more than one operations inside a bracket, they are carried out first, following rules 2 and 3.