Mathematics Lab Activity-05 Class X | Arithmatic Progression

 Mathematics Lab Activity-05 Class X

Mathematics Laboratory Activities on Arithmatic Progression for class X students with complete observation tables strictly according to the CBSE syllabus.

Chapter - 05 

arithmatic progression

Activity - 05

Objective

To find the sum of first n natural numbers.

Material Required

Cardboard, coloured papers, white paper, cutter, adhesive.

Procedure

1. Take a rectangular cardboard of a convenient size and paste a colored paper on it. Draw a rectangle ABCD of length 11 units and breadth 10 units.

2. Divide this rectangle into unit squares as shown in Fig. 1.

3. Starting from upper left-most corner, color one square, 2 squares and so on as shown in the figure 1.

Figure 1

4. The pink color region looks like a stair case.

5. Length of 1st stair is 1 unit, length of 2nd stair is 2 units, length of 3rd stair 3 units, and so on, length of 10th stair is 10 units.

6. These lengths give a pattern 1, 2, 3, 4, ..., 10,

7. which is an AP with first term 1 and common difference 1.

Observations

1. Sum of first ten terms

    = 1 + 2 + 3 + ... +10 = 55 .........(1)

Area of the shaded (Pink) region 

                    = 1/2 (area of rectangle ABCD)

                = 1/2(10 x 11)

                = 55

It is same as obtained in eqn 1

This shows that the sum of the first 10 natural numbers is

              

             

This can be generalised to find the sum of first n natural numbers as 

       

2. For n = 4, S₄ = 10

3. For n = 12, S₁₂ = 78

4. For n = 50, S₅₀ = 1275

5. For n = 100, S₁₀₀ = 5050 

Result

Sum of first n natural number is given by 

        

APPLICATION

Result (2) may be used to find the sum of first n terms of the list of numbers:

1. 1², 2², 3², ...

2. 1³, 2³, 3³, ...