Math Labs with Activity - Sum of the First n Natural Numbers

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

1. A sheet of white paper
2. A geometry box
3. A tube of glue
4. A long, colored paper tape of uniform width (say, 1 unit)
5. 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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