Mathematics Lab Activity-14 Class XI

Mathematics Laboratory Activities on Sequence & Seriesfor class XI students Non-Medical. These activities are according to the CBSE syllabus

Chapter - 8 Sequence & Series
Activity - 14

Objective

To demonstrate that the Arithmetic mean of two different positive numbers is always greater than the Geometric mean

Material Required

Colored chart paper, sketch pens, ruler, cutter.

Theory

Arithmetic Mean (A. M.): For given two numbers a and b, we can insert a number A between a and b such that a, A, b are in A.P.(Arithmetic progression). Such a number A is called the arithmetic mean(A.M.) of the given numbers a and b.

As a, A, b are in A.P. therefore,

A – a = b - A
2A = a + b

Geometric Mean(G.M.)

Geometric mean of two positive numbers a and b is defined as the number .

Procedure

1. Cut four rectangular pieces (I, II, III and IV) of dimensions a x b (a > b) from the given chart paper.

2. Arrange all these pieces as shown in figure 14.1

Figure 14.1

3. ABCD is a square of side (a + b) units.

Area of square ABCD = (a + b)² square units.

4. Area of four rectangular pieces(I, II, III, IV) 

     = (a x b) x 4 square units.

     = 4ab square units.

5. PQRS is a square of side (a - b) units.

Area of  square PQRS = (a - b)² square units.

Observations

1. Area of square ABCD = Area of four rectangular pieces+ area of square PQRS

i.e.  (a + b)² = 4ab + (a - b)²

Subtracting (a - b)² from the right hand side

⇒  (a + b)²  > 4ab

⇒   (a + b) >

⇒   (a + b) >  

⇒     

⇒  A.M. > G.M.

2. Take a = 6 cm and b = 4 cm

AB = a + b = 6 cm + 4 cm = 10 cm

Area of square ABCD = (a + b)² = (10cm)² = 100cm²

Area of each rectangle = a x b = 6cm x 4cm = 24 cm².

Area of square PQRS  = (a – b)² = (6 - 4)² = 2² = 4 cm².

3. Area of square ABCD = 4 x Area of rectangle  + Area of square  PQRS

⇒100 = 4 x 24 + 4
⇒100 = 96 + 4
⇒100 > 96
⇒ A.M. > G.M.

Result

The arithmetic mean of two different positive numbers is always greater than their geometric mean.

Applications

This activity can be used to prove that the arithmetic mean of two different positive numbers is always greater than their geometric mean.

VIVA – VOICE

Q. 1. What is the arithmetic mean of two real positive numbers a and b ?

Ans.

Q. 2. What is geometric mean of two real positive numbers a and b ?

Ans.  

Q. 3. Which is greater, A.M. or G.M. of two real positive numbers?

Arithmetic Mean (A.M.) is greater than Geometric Mean(G.M.)

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