Note:

1. If there are Geometric Means inserted between and , then

2. Geometric mean between and is

Question 1: Insert 6 geometric means between and .

Answer:

Let be the GMs inserted between and .

( As geometric means are being inserted).

. Therefore,

Hence the 6 geometric means between and are

Question 2: Insert 5 geometric means between and .

Answer:

Let be the GMs inserted between and .

( As geometric means are being inserted).

. Therefore,

Hence the 5 geometric means between and are

Question 3: Insert 5 geometric means between and .

Answer:

Let be the GMs inserted between and .

( As geometric means are being inserted).

. Therefore,

Hence the 5 geometric means between and are

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

i) ii) iii)

Answer:

i)

Geometric mean between

ii)

Geometric mean between

iii)

Geometric mean between

Question 5: If is the G.M. of and , find

Answer:

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

Answer:

Given A.M. … … … … … i)

Also GM … … … … … ii)

Solving i) and ii) we get

or

When

When

Hence the two numbers are and

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

Answer:

Let the two roots of the quadratic equation be and

… … … … … i)

Also … … … … … ii)

We know the quadratic equation having roots and is given by

Question 8: The sum of two numbers is times their geometric mean, show that the numbers are in the ratio .

Answer:

Let the two numbers be and

Given

Applying componendo and dividendo

Applying componendo and dividendo once again

. Hence proved.

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

Answer:

Let the two roots of the quadratic equation be and

Given AM … … … … … i)

Also GM

Therefore the quadratic equation will be

Question 10: If AM and GM of two positive numbers are respectively, find the numbers.

Answer:

Given AM … … … … … i)

Also GM … … … … … ii)

Solving i) and ii) we get

When

When

Hence the two numbers are and .

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

Answer:

Let be the GMs inserted between and

Let be the common ratio.

Let be the GM between and

Now Given

. Hence proved.

Question 12: If the A.M. of two positive numbers and is twice their geometric mean. Prove that:

Answer:

Let the two numbers be and

Given A.M. = 2 G.M.

Given

Applying componendo and dividendo

Applying componendo and dividendo once again

. Hence proved.

Question 13: If one A.M., and two geometric means and inserted between any two positive numbers, show that

Answer:

Let the two numbers be and

Let and be two GM between and

To prove :

LHS

RHS