Page 35 - Mathematics Class - X

3. Divide this rectangle into unit squares (1 cm × 1 cm).

4. Fix a thermocol ball in each square using a pin as shown in the Fig. (a).

5. Enclose ( ) the balls as shown in the Fig. (a)

6. Starting from the uppermost left corner (Fig. (a))

(a) The number of balls in first enclosure = 2 (= 1 × 2)

(b) The number of balls in first two enclosures = 2 + 4 = 6 ( = 2 × 3)

(c) The number of balls in first three enclosures = 2 + 4 + 6 = 12 (= 3 × 4)

(d) The number of balls in first six enclosures = 2 + 4 + 6 + 8 + 10 + 12 = 42 (= 6 × 7)

(e) The number of balls in first ten enclosures = 2 + 4 + 6 + 8 + ... + 20 = 110 (= 10 × 11)
7. This gives the sum of first ten even natural numbers.

OBSERVATIONS

For n = 4 in equation (i), S = ____________
n
For n = 6 in equation (i), S = ____________
n
For n = 40 in equation (i), S = ____________
n
For n = 70 in equation (i), S = ____________
n
For n = 100 in equation (i), S = ____________
n

INFERENCE
The sum of first n even natural numbers,

S = 2 + 4 + 6 + ... + 2n = n × (n + 1)
n

EXTENDED TASK

1. Find the sum of first 40 even natural numbers.
2. Find the sum of (22 + 24 ...+ 100) by using this activity.

APPLICATION

The formula S = n (n + 1) is useful in finding out the sum of the first n even natural numbers.
n

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