Class 11 RD Sharma Solutions – Chapter 20 Geometric Progressions- Exercise 20.6

Question 1: Insert 6 geometric means between 27 and 1/81.

Solution:

Let the six geometric means be A₁, A₂, A₃, A₄, A₅, A₆.

Now, these 6 terms are to be added between 27 and 1/81.

So the G.P. becomes, 27, A₁, A₂, A₃, A₄, A₅, A₆,1/81 with first term(a) = 27, number of terms(n) = 8 and 8th term(a₈) = 1/81.

We know nth term of a G.P. is given by a_n = ar^n-1, where r is the common ratio.

=> a₈ = 1/81

=> 27(r)^8-1 = 1/81

=> r⁷ = 1/(81×27)

=> r⁷ = (1/3)⁷

=> r = 1/3

So, geometric means are:

A₁ = ar = 27(1/3) = 9

A₂ = ar² = 27(1/3)² = 3

A₃ = ar³ = 27(1/3)³ = 1

A₄ = ar⁴ = 27(1/3)⁴ = 1/3

A₅ = ar⁵ = 27(1/3)⁵ = 1/9

A₆ = ar⁶ = 27(1/3)⁶ = 1/27

Therefore, the 6 geometric means between 27 and 1/81 are 9, 3, 1, 1/3, 1/9, 1/27. 

Question 2. Insert 5 geometric means between 16 and 1/4.

Solution:

Let the five geometric means be A₁, A₂, A₃, A₄, A₅.

Now, these 5 terms are to be added between 16 and 1/4.

So the G.P. becomes, 16, A₁, A₂, A₃, A₄, A₅,1/4 with first term(a) = 16, number of terms(n) = 7 and 7th term(a₇) = 1/4.

We know nth term of a G.P. is given by a_n = ar^n-1, where r is the common ratio.

=> a₇ = 1/4

=> 16(r^7-1) = 1/4

=> r⁶ = 1/64

=> r⁶ = (1/2)⁶

=> r = 1/2

So, geometric means are:

A₁ = ar = 16(1/2) = 8

A₂ = ar² = 16(1/2)² = 4

A₃ = ar³ = 16(1/2)³ = 2

A₄ = ar⁴ = 16(1/2)⁴ = 1

A₅ = ar⁵ = 16(1/2)⁵ = 1/2

Therefore, the 5 geometric means between 16 and 1/4 are 8, 4, 2, 1, 1/2.

Question 3. Insert 5 geometric means between 32/9 and 81/2.

Solution:

Let the five geometric means be A₁, A₂, A₃, A₄, A₅.

Now, these 5 terms are to be added between 32/9 and 81/2.

So the G.P. becomes, 32/9, A₁, A₂, A₃, A₄, A₅, 81/2 with first term(a) = 32/9, number of terms(n) = 7 and 7th term(a₇) = 81/2.

We know nth term of a G.P. is given by a_n = ar^n-1, where r is the common ratio.

=> a₇ = 81/2

=> (32/9)(r^7-1) = 81/2

=> r⁶ = (81×9)/(2×32)

=> r⁶ = (3/2)⁶

=> r = 3/2

So, geometric means are:

A₁ = ar = (32/9)×(3/2) = 16/3

A₂ = ar² = (32/9)×(3/2)² = 8

A₃ = ar³ = (32/9)×(3/2)³ = 12

A₄ = ar⁴ = (32/9)×(3/2)⁴ = 18

A₅ = ar⁵ = (32/9)×(3/2)⁵ = 27

Therefore, the 5 geometric means between 32/9 and 81/2 are 16/3, 8, 12, 18, 27.

Question 4. Find the geometric means of the following pairs of numbers:

(i) 2 and 8

(ii) a³b and ab³

(iii) –8 and –2

Solution:

We know geometric mean between two numbers, a and b is given by .

(i) 2 and 8

Here, a = 2 and b = 8

So, G.M. = 

= 

= 4

(ii) a³b and ab³

Here, a = a³b and b = ab³

So, G.M. = 

= 

= a²b²

(iii) –8 and –2

Here, a = –8 and b = –2

G.M. = 

= 

= 4

Question 5. If a is the G.M. of 2 and 1/4 find a.

Solution:

We know geometric mean between two numbers, a and b is given by .

According to the question,

a = 

= 

= 

Therefore, the value of a is .

Question 6. Find the two numbers whose A.M. is 25 and GM is 20.

Solution:

We know geometric mean between two numbers, a and b is given by  and arithmetic mean between two numbers, a and b is given by (a+b)/2.

According to the question,

=>  = 20 ……. (1)

And

=> (a+b)/2 = 25

=> a+b = 50

=> b = 50–a ……. (2)

From (1) and (2), we get,

=>  = 20

Squaring both sides, we get,

=> a(50 –a) = 400

=> a² – 50a + 400 = 0

=> a²– 40a–10a+400 = 0

=> a(a– 40) – 10(a– 40) = 0

=> (a– 40) (a– 10) = 0

=> a = 40 or a = 10

Putting these in (2) we get,

When a = 40, then b = 10 and 

When a = 10, then b = 40.

Therefore, the numbers are 10 and 40.

Question 7. Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

Solution:

Suppose the roots of the quadratic equation are a and b.

We know geometric mean between two numbers, a and b is given by  and arithmetic mean between two numbers, a and b is given by (a+b)/2.

According to the question,

A.M. of roots = (a+b)/2 = A

a + b = 2A ….. (1) 

And G.M. of roots =  = G

ab = G²  … (2)

Now, we know a quadratic equation in x with roots a and b is given by,

x² – (a+b)x + (ab) = 0

From (1) and (2), we get,

x² – 2Ax + G² = 0 

Therefore, the required quadratic equation is x² – 2Ax + G² = 0.

Question 8. The sum of two numbers is 6 times their geometric means. Show that the numbers are in the ratio 

Solution:

Let the two numbers be a and b. We know geometric mean between two numbers, a and b is given by .

According to the question,

=> a+b = 6

=>  = 

Applying Componendo and Dividendo on both sides, we get,

=>  = 

=>  = 

=>  = 

By again applying Componendo and Dividendo on both sides, we get,

=>  = 

=>  = 

=>  = 

Squaring both sides, we get

=>  = 

=>  = 

=>  = 

Hence proved.

Question 9. If AM and GM of roots of a quadratic equation are 8 and 5 respectively, then obtain the quadratic equation.

Solution:

Suppose the roots of the quadratic equation are a and b.

We know geometric mean between two numbers, a and b is given by  and arithmetic mean between two numbers, a and b is given by (a+b)/2.

According to the question,

A.M. of roots = (a+b)/2 = 8

a+b = 16 ….. (1)

And G.M. of roots =  = 5

ab = 25  .… (2)

Now, let the quadratic equation with roots a and b is given by,

x² – (a+b)x + (ab) = 0

From (1) and (2), we get,

x² – 16x + 25 = 0

Therefore, the required quadratic equation is x² – 16x + 25 = 0.

Question 10. If AM and GM of the two positive numbers a and b are 10 and 8 respectively. Find the numbers.

Solution:

We know geometric mean between two numbers, a and b is given by  and arithmetic mean between two numbers, a and b is given by (a+b)/2.

According to the question,

=>  = 8 ……. (1)

And,

=> (a+b)/2 = 10

=> a+b = 20

=> b = 20–a ……. (2)

From (1) and (2), we get,

=>  = 8

Squaring both sides, we get,

=> a(20–a) = 64

=> a²–20a+64 = 0

=> a²–16a–4a+64 = 0

=> a(a–16) – 4(a–16) = 0

=> (a–4) (a–16) = 0

=> a = 4 or a = 16

Putting a = 4 in (2), we get b = 16. And,

Putting a = 16 in (2), we get b = 4.

Therefore, the numbers are 4 and 16.

Question 11. Prove that the product of n geometric means between two quantities is equal to the nth power of geometric mean of those two quantities.

Solution:

Suppose we have a GP with first term a, common ratio r and number of terms n.

We have to add these n terms of GP between two quantities such that the GP remains maintained. So total number of terms become (n+2).

We know nth term of a G.P. is given by a_n = ar^n-1.

So, the last term of the GP ,i.e., (n+2)^th term will be, a_n+2 = ar^n+2-1 = arⁿ⁺¹  

We know geometric mean between two numbers, a and b is given by .

The GM of a and arⁿ⁺¹ will be, G₁ =  =   

Hence, L.H.S. =  = 

Now, R.H.S. = Product of n geometric means between these two quantities, G₂ = ar × ar² × . . . . × arⁿ

= 

= 

= 

= 

= L.H.S.

Hence, Proved.

Question 12: If the AM of two positive numbers a and b (a>b) is twice their geometric mean. Prove that a:b = 

Solution:

We know geometric mean between two numbers, a and b is given by  and arithmetic mean between two numbers, a and b is given by (a+b)/2.

According to the question,

AM = 2(GM)

=>  = 

=>  = 

Applying Componendo and Dividendo on both sides, we get,

=>  = 

=>  = 

=> = 

By again applying Componendo and Dividendo on both sides, we get,

=> = 

=> = 

=>  = 

Squaring both sides, we get

=>  = 

=>  = 

=>  = 

=>  = 

Hence, proved.

Question 13. If one AM, A, and two geometric means G₁ and G₂ are inserted between any two positive numbers, show that 

Solution:

Let the two positive numbers be a and b.

Now value of one AM between a and b, A = (a+b)/2.

So, 2A = a+b . . . . (1) 

If we add two geometric means between a and b, the GP becomes a,G₁,G₂,b.

Now, we know b = ar^4-1, where r is the common ratio.

=> r³ = 

=> r = 

So, G₁ = ar =  = 

G₂ = ar² =  = 

Now, L.H.S. = 

= 

= ab⁰ + a⁰b

= a+b

= 2A [From (1)]

= R.H.S.

Hence, proved.