Class X Chapter - 05
Mathematics Laboratory Activities on Arithmatic Progression for class X students with complete observation tables strictly according to the CBSE syllabus.
arithmatic progression
Activity - 07
Objective
To find the sum of the first n-even natural numbers.
Material Required
Cardboard, thermocol balls, pins, pencil, ruler, white paper, chart paper, adhesive.
Procedure1. Take a piece of cardboard of a convenient size and paste a white paper on it.
2. Draw a rectangle of suitable size on it (10 cm × 11 cm).
3. Divide this rectangle into unit squares.
4. Fix a thermocol ball in each square using a pin as shown in the Fig. 1.
5. Enclose the balls as shown in the figure.
6. Starting from the uppermost left corner.
the number of balls in first enclosure = 2 = (1 × 2),
the number of balls in first two enclosures = 2 + 4 = 6 = (2 × 3),
the number of balls in first three enclosures = 2 + 4 + 6 = 12 = (3 × 4),
the number of balls in first six enclosures = 2 + 4 + 6 + 8 + 10 + 12 = 42 = (6 × 7)
the number of balls in first ten enclosures = 2 + 4 + 6 + 8 + ... + 20 = 110 = (10 × 11)
Figure 1
This gives the sum of first ten even natural numbers.
This result can be generalised for the sum of first n even natural numbers as
Sn = 2 + 4 + 6 + ... + 2n
Sn = n × (n + 1)
Observations
For n = 4 in (1), Sn = 4 × 5 = 20
For n = 7 in (1), Sn = 7 × 8 = 56
For n = 40 in (1), Sn = 40 × 41 = 1640
For n = 70 in (1), Sn = 70 × 71 = 4970
For n = 100 in (1), Sn = 100 × 101 = 10100
Result
Sum of first n even natural number is given by Sn = n (n + 1), where n is the number of terms.
Applications
The formula Sn = n (n + 1) is useful in finding out the sum of the first n even natural numbers.
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