Mathematics Lab Activity-06 Class X

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

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.

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.

Figure 1

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

(blue colour) = 1 (=1²).

Observations

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

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

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

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 …………… (1)

2. For n = 4, S₄ = 4² = 16

3. For n = 12, S₁₂ = 12² = 144

4. For n = 50, S₅₀ = 50² = 2500

5. 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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