Math, asked by firdous199507, 5 months ago

Activity: To find the sum of first n natural numbers.

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Answers

Answered by bijayjames291

Answer:

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.

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 =

Math Labs with Activity – Sum of the First n Natural Numbers 6for n = 10.

Result

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

Math Labs with Activity – Sum of the First n Natural Numbers 7

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

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Filed Under: CBSE

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Activity: To find the sum of first n natural numbers.please don't spam it because spam answer will be reported......​

Answers

Activity: To find the sum of first n natural numbers.

please don't spam it because spam answer will be reported......