Arithmetic progression

Colored paper, Pair of scissors, Geometry box, Fevicol, Sketch paper, One squared paper.

- Take a given sequence of numbers say a
_{1}, a_{2}, a_{3}.... - Cut a rectangular strip from a coloured paper of width k = 1 cm (say) and length a
_{1}cm. - Repeat this procedure by cutting rectangular strips of the same width k = 1cm and lengths a
_{2}, a_{3}, a_{4}.. cm. - Take 1 cm squared paper and paste the rectangular strips adjacent to each other in order.

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**[A]Let the sequence be 1, 4, 7, 10, ....**

- Take strips of lengths 1 cm, 4 cm, 7 cm and 10 cm, all of the same width say 1 cm.
- Arrange the strips in order as shown in Fig 1(a).
- Observe that the adjoining strips have a common difference in heights. (In this example it is 3 cm.)

Figure 1(a)

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**[B]Let another sequence be 1, 4, 6, 9, ...**

- Take strips of lengths 1 cm, 4 cm, 6 cm and 9 cm all of the same width say 1 cm.
- Arrange them in an order as shown in Fig 1(b).
- Observe that in this case the adjoining strips do not have the same difference in heights.

Figure 1(b)

- Click on each box provided in workbench area. Here, we will be generating sequence of 1,4,7,10....
- Click on next button below.
- Click on each box provided in workbench area. Here, we will be generating sequence of 1,4,6,9....
- Click on next button below.

So, it is observed that if the given sequence is an arithmetic progression, a ladder is formed in which the difference between the heights of adjoining steps is constant. If the sequence is not an arithmetic progression, a ladder is formed in which the difference between adjoining steps is not constant.

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