**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**

- A sheet of white paper
- 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.

- Number of squares marked A = 1.

Number of squares marked B = 3.

Total number = 4 = 2².

1 + 3 = 2². - 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.

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