Mathematics Lab Activity-06 Class X | Arithmatic Progression

Chapter - 05   
Mathematics Lab Activity Class X

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

arithmatic progression

Activity - 06

Objective

To find the sum of the first n odd natural numbers.

Material Required

Cardboard, thermocol balls, pins, pencil, ruler, adhesive, white paper.

Procedure

1. Take a piece of cardboard of a convenient size and paste a white paper on it.

2. Draw a square of suitable size on it (10 cm × 10 cm).

3. Divide this square into unit squares.

Figure 1

4. Fix a thermocol ball in each square with the help of a pin as shown in Fig. 1.

5. Enclose the balls as shown in the figure.

6. Starting from the uppermost right corner, the number of balls in first enclosure

(blue colour) = 1 = (1²).

7. The number of balls in first 2 enclosures = 1 + 3 = 4 = (2²).

8. The number of balls in first 3 enclosures = 1 + 3 + 5 = 9 = (3²).

9. The number of balls in first 10 enclosures = 1 + 3 + 5 + ... + 19 = 100 = (10²).

10. This gives the sum of first ten odd natural numbers. This result can be generalised for the sum of first n odd numbers as:

S_n = 1 + 3 + ......... + (2n – 1) = n²

Observations

S_n = 1 + 3 + ......... + (2n – 1) = n²      …………… (1)

For n = 4, S₄             =  4²    = 16

For n = 12, S₁₂         =  12²  = 144

For n = 50, S₅₀         =  50²  = 2500

For n = 100, S₁₀₀ =  100² = 10000

Result

Sum of first n odd natural number is given by n² where n is the number of terms.    

Applications

The activity is useful in determining formula for the sum of the first n odd natural numbers.

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