Arithmetic Progression

 

Arithmetic Progression I

Objective
To verify that the given sequence is an arithmetic progression by paper cutting and pasting method

Prerequisite Knowledge
Understanding the concept of an arithmetic progression.

Arithmetic Progression:
A sequence is known as an arithmetic progression (sequence) if the difference between the term and its predecessor always remains constant.

Materials Required
Coloured papers, a pair of scissors, fevicol, geometry box, sketch pens, drawing sheets.

Procedure

1.  Take a given sequence of numbers say A1, A2,  A,…
2.  Cut a rectangular strip from coloured paper of width 1 cm and length A, cm.
3. Repeat the procedure by cutting rectangular strips of same width 1 cm and lengths A,, A3…cm.
4.  Take a graph paper and paste these rectangular strips adjacent to each other in order on graph paper.

[A] Consider a sequence 1, 4, 7, 10, 13.

1.  Take different colour strips of lengths 1 cm, 4 cm, 7 cm, 10 cm, 13 cm and all of the same width 1 cm
   (say).
2.  Arrange and paste these strips in order on a graph paper as shown in fig. (i).

[B] Consider a sequence 1, 4, 8, 10, 11.

1.  Take different colour strips of lengths 1 cm, 4 cm, 8 cm, 10 cm, 11 cm and all of the same width 1 cm (say).
2.  Arrange and paste these strips in order on a graph paper as shown in fig. (ii).

Observation
We observe from fig(i) that the adjoining strips have a common difference in heights i.e. 3 cm and a ladder is formed in which the adjoining steps are constant. Hence it is an arithmetic progression.
In fig (ii) the adjoining strips don’t have a common difference in heights and thus the adjoining steps of ladder are not constant. Hence it is not an arithmetic progression.

Result
Sequence [A] is an AP because common difference between the term and its predecessor remains constant.
Sequence [B] is not an AP because common difference between the term and its predecessor does not remain constant.