**Math Labs with Activity – Sequence of Numbers is an Arithmetic Progression (AP)**

**OBJECTIVE**

To verify using a graphical method that a given sequence of numbers is an arithmetic progression (AP)

**Materials Required**

- Two sheets of graph paper
- A ruler
- A pencil
- A long, colored paper tape of uniform width (say, 1 unit)
- A tube of glue

**Theory**

A sequence of numbers in which every term except the first term is obtained by adding a fixed number to its preceding term is called an arithmetic progression (AP).

**Procedure**

**Step 1:** Mark both the sheets of graph paper with squares (each side = 1 unit).

**Step 2:** Mark x- and y-axes on each sheet of graph paper.

**Step 3:** We shall first test if the sequence 2,5,8,11,14,… is an AP.

**Step 4:** Cut the paper tape in rectangular strips of lengths 2 units, 5 units, 8 units, 11 units, 14 units,….

**Step 5:** Using the x-axis as the base, paste the strips on graph paper I sequentially (as shown in Figure 2.1). Record your observations in the first observation table.

**Step 6:** We shall now test if the sequence 2,6,9,13,16,… is an AP.

**Step 7:** Cut the paper tape in rectangular strips of lengths 2 units, 6 units, 9 units, 13 units, 16 units,….

**Step 8:** Using the x-axis as the base, paste the strips on graph paper II sequentially (as shown in Figure 2.2). Record your observations in the second observation table.

**Observations**

**I. For Figure 2.1**

**Conclusions**

- The sequence 2, 5, 8, 11, 14, … forms a uniform ladder having equal steps (as shown in Figure 2.1) and has a common difference d = 3 units (see the first observation table). Hence, this sequence is an AP.
- The sequence 2,6,9,13,16,… forms a ladder but of unequal steps (as shown in Figure 2.2) and the common difference (d) does not exist (see the second observation table). Hence, this sequence is not an AP.

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