**Math Labs with Activity – Sum of the First n Natural Numbers**

**OBJECTIVE**

To verify by a graphical method that the sum of the first ti natural numbers is

**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**

The sum of the first n terms of an AP is given by

….(i)

Taking a = 1 and d = 1, the AP becomes 1,2, 3, 4, …

**∴** putting a =1 and d = 1 in equation (i), we get

i.e., the sum of the first n natural numbers =

**Procedure**

**Step 1:** We shall verify the formula for the first 10 natural numbers, i.e., for n = 10.

**Step 2:** Draw horizontal and vertical lines on the sheet of white paper to mark squares, each of size 1 unit x 1 unit.

**Step 3:** Cut the colored paper tape in 10 rectangular strips of lengths 1 unit, 2 units, 3 units, …, 10 units.

**Step 4:** Paste these strips on the grid sequentially as shown in Figure 4.1.

**Step 5:** Complete the rectangle ABCD so that AB = 11 units and BC = 10 units.

**Step 6:** Cut the portion of the rectangle ABCD which is covered by the colored paper strips. We find that this portion completely covers the remaining portion of the rectangle ABCD.

**Observations and Calculations**

Area of the complete rectangle ABCD =(10 x 11) square units.

**∴** area of the grid covered by the paper strips = ½ (10 x 11) square units.

Also, the area of each square of the grid =1 x 1 = 1 square unit.

Total number of squares covered by the paper tape = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.

**∴** area of the grid covered by the paper tape

= (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) square units. … (ii)

From (i) and (ii), we get

1 + 2+ 3 + 4+5 + 6+7 + 8 + 9 +10 = ½(10 x 11)

i.e., 1 + 2+ 3+…+n =

for n = 10.

**Result**

It is verified for n = 10 that the sum of the first n natural numbers is

**Remarks:** The students must try to verify this identity for various other values of n.

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