Page 37 - Mathematics Class - X

ACTIVITY 2.8

OBJECTIVE
To establish a formula for the sum of first n terms of an Arithmetic Progression.

MATERIALS REQUIRED

Cardboard White paper
Coloured drawing sheets Cutter and adhesive

PRE-REQUISITE KNOWLEDGE

Knowledge of Arithmetic Progression

THEORY
1. For concept of AP refer to Activity 2.4.

2. Consider an AP, a, a + d, a + 2d,..., a + (n – 1)d
where, nth term of the series is [a + (n – 1)d].

Let S denotes the sum of first n terms of this AP,
\ S = a + (a + d) + (a + 2d) +...+ [a + (n – 1)d] ...(i)

Rewriting the terms in reverse order, we get
S = [a + (n – 1)d] + [a + (n – 2)d] +...+ (a + d) + a ...(ii)

On adding Eqs. (i) and (ii), term wise, we get
2S = [2a + (n – 1)d] + [2a + (n – 1)d] +...+ [2a + (n – 1)d] n times

or 2S = n [2a + (n – 1)d] [since, there are n terms]
n
or S = [2a + (n – 1)d]
2
So, the sum of the first n terms of an AP is given by
n
S = [2a + (n − 1) ] d
2

PROCEDURE

1. Take a rectangular cardboard and paste a white paper on it.

2. Draw a rectangle ABCD of length (2a + 9d) units and breadth 10 units.
3. Make some rectangular strips of equal length a units and breadth one unit and some strips of length d units
and breadth 1 unit using coloured drawing sheets.

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