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

$=\frac{1}{2}(10\times 11)$

$=\frac{1}{2}\times 10\times(10+1)$

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

$S_{n}=\frac{1}{2}n(n+1)$

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

$S_{n}=\frac{1}{2}n(n+1)$

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, ...

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