Question

Calculate the sum of first n natural numbers given by your teacher through the following activity. For example to find the sum of first four natural numbers, take a square-grid piece of paper of 4×5 squares.Then cut it as shown in the figure Hence verify the formula Sn= (n+1)÷2

Answer 1

Sₙ = n(n+1)/2

Step-by-step explanation:

Let shade the 1 , 2 , 3  , 4

Now Cut the Shaded area

and match with unshaded area  ( rotate 180°)

Shaded Area = unshaded area

Shaded + Unshaded = 4 * 5

Shaded = 4 * 5/2

=> Sₙ = n(n+1)/2

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In an A.P. 1st term is 1 and the last term is 20. The sum of all terms is ...

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Answer 2

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Step-by-step explanation:

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

Math Labs with Activity – Sum of the First n Natural Numbers 1

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

Math Labs with Activity – Sum of the First n Natural Numbers 2 ….(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

Math Labs with Activity – Sum of the First n Natural Numbers 3

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

Math Labs with Activity – Sum of the First n Natural Numbers 4

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.

Math Labs with Activity – Sum of the First n Natural Numbers 5

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.