Page 34 - Mathematics Class - X

ACTIVITY 2.7

OBJECTIVE
To find the sum of first n even natural numbers.

MATERIALS REQUIRED

Cardboard Pencil Chart paper
Thermocol balls Ruler Adhesive
Pins White sheet

PRE-REQUISITE KNOWLEDGE
1. Concept of natural numbers 3. Concept of series formed by natural numbers
2. Concept of Arithmetic Progression (AP)

THEORY
1. The natural numbers which are completely divisible by 2 are called even natural numbers, e.g. 2, 4, 6, 8, ... .

2. In 2, 4, 6, 8, ..., the difference between any two consecutive numbers are same, i.e. 2. So, it is an AP.
nth term of an AP is T = a + (n – 1)d
n
Here, a = 2, d = 4 – 2 = 6 – 4 = 8 – 6 = 2
∴ T = 2 + (n – 1) 2 = 2 + 2n – 2 = 2n
n
So, the series can be written as 2, 4, 6, 8, ..., 2n.

We know that the sum of first n terms of a given arithmetic progression is
n
S = [2a + (n − 1) ] d
n
2
For the series 2, 4, 6, 8, ..., 2n; a = 2, d = 2
n n 2n
∴ S = [2 2×+ (n − 1) 2]× = [4 + 2n − 2] = (n + 1) = ( nn + 1) ....... (i)
n 2 2 2

PROCEDURE
1. Take a piece of cardboard and paste a white sheet on it.
2. Let us find the sum of first 10 even natural numbers,
i.e. 2, 4, 6, ..., 20. So, draw a rectangle of suitable size on
it (10 cm × 11 cm).

Fig. (a)

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