**Math Labs with Activity – Sum of the First n Terms of an AP**

**OBJECTIVE**

To verify that the sum of the first n terms of an arithmetic progression where a is the first term and d is the common difference is given by

**Materials Required**

- A sheet of white paper
- A geometry box
- A tube of glue
- A long, colored paper tape of uniform width (say, 1 unit)
- A pencil

**Theory**

If a is the first term, d the common difference and l the nth term of an AP then

l=a + (n-1)d. … (i)

Now, the sum of n terms of an AP is given by

[using equation (i)].

**Procedure**

**Step 1:** We shall verify the above formula for a general AP having the first term a and the common difference d for n = 10.

**Step 2:** Draw horizontal lines on the sheet of paper with a distance of 1 unit between two consecutive lines.

**Step 3:** Cut 10 small rectangular strips from the coloured paper tape, each of the same length (say, a units).

**Step 4:** Cut 45 other small rectangular strips from the paper tape, each of the same length (say, d units).

**Step 5:** Paste both types of strips on the white paper along the horizontal lines so as to obtain rectangles of lengths a,a + d,a + 2d,…,a+9d arranged sequentially, as shown in Figure 3.1.

**Step 6:** Extend the line DE to C by a units to construct the rectangle ABCD (as shown in Figure 3.1).

**Step 7:** Cut the portion of the rectangle ABCD which is covered with the coloured paper tape. We find that this portion completely covers the remaining portion of the rectangle ABCD.

**Observations and Calculations**

- The length of the rectangle ABCD = (a + 9d) + a = 2a + 9d and the breadth of the rectangle ABCD =10×1=10 units.

∴ the area of the rectangle ABCD = 10(2a + 9d) units² … (ii) - Area of the portion of the rectangle ABCD covered with coloured strips of paper tape = sum of the areas of the 10 rectangles

= (a x 1) + [(a + d) x 1] + [(a + 2d) x 1]+…+ [(a + 9d) x 1]

= a+(a + d)+(a + 2d) +…+ (a + 9d). … (iii) - Area of the portion of the rectangle ABCD covered by the coloured strips = ½ (area of the rectangle ABCD).

a + (a + d)+(a + 2d) +…+(a + 9d) = 10/2 (2a + 9d) [using equations (ii) and (iii)]

i.e., a + (a + d) + (a + 2d) +…+ [a + (n -1 )d] = n/2 [2a+(n -1 )d] for n = 10.

**Result**

It is verified for n = 10 that the sum of the first n terms of an AP is given by

**Remarks:**

The students shall apply the above method of verification for various values of n, taking different values of a and d as well.

Math Labs with ActivityMath LabsScience Practical SkillsScience Labs

Math Labs with ActivityMath LabsScience Practical SkillsScience Labs