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

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

OBJECTIVE

To verily by a graphical method that the sum of the first n odd natural numbers is equal to n1

Materials Required

1. A sheet of white paper
2. A geometry box

Theory
The sum of the first n odd natural numbers is given by
Σ(2n – 1) =1 + 3 + 5 + … + (2n – 1) = n².

Procedure
Step 1: Make a square of side 10 cm on the sheet of paper.
Step 2: Draw horizontal and vertical lines in this square so as to make 100 smaller squares within this square. Then, each small square will have a side = 1 cm.
Step 3: Mark the small squares with letters as shown in Figure 5.1.

Observations and Calculations
From Figure 5.1, we observe the following.

1. Number of squares marked A = 1.
   Number of squares marked B = 3.
   Total number = 4 = 2².
   1 + 3 = 2².
2. Number of squares marked A = 1.
   Number of squares marked B = 3.
   Number of squares marked C = 5.
   Total number = 9 = 3².
   1 + 3+5 = 3².
   Proceeding similarly, we get 1 + 3 + 5 +.. ,…+ 19=10².

Result
It is verified for n = 1, 2, 3,…, 10 that the sum of the first n odd natural numbers is equal ton2.
In other words, Σ(2n – 1) =1 + 3+ 5 + …. + (2n – 1) =n².

Math Labs with ActivityMath LabsScience Practical SkillsScience Labs