Question 1: Find the . between:

(i) and (ii) and (iii) and

Answer:

i) Let be the Arithmetic Mean of and .

Therefore are in A.P.

Hence the Arithmetic Mean of and is .

ii) Let be the Arithmetic Mean of and .

Therefore are in A.P.

Hence the Arithmetic Mean of and is .

iii) Let be the Arithmetic Mean of and .

Therefore are in A.P.

Hence the Arithmetic Mean of and is .

Question 2: Insert between and .

Answer:

Let be the 4 A.M.s between and .

Therefore are in A.P.

We know

Therefore

Therefore the 4 A.M.s between and are

Question 3: Insert between and .

Answer:

Let be the 7 A.M.s between and .

Therefore are in A.P.

We know

Therefore

Therefore the 4 A.M.s between and are

Question 4: Insert between and

Answer:

Let be the 6 A.M.s between and .

Therefore are in A.P.

We know

Therefore

Therefore the 6 A.M.s between and are

Question 5: There are between and . The ratio of the last mean to the first mean is . Find the value of .

Answer:

Let be the arithmetic means between and

and

We know

Question 6: Insert , between and in such a way that is . Find the number of .

Answer:

Let be the arithmetic means between and

Given

Therefore there are A.M.s

Question 7: If are inserted between two numbers , prove that the sum of the mean equidistant from the beginning and the end is constant.

Answer:

Let be the arithmetic means between and

and

A.M. between and which is constant.

Question 8: If are in A.P. and is the of and and is the of and , then prove that the of and is .

Answer:

Given are in A.P. Therefore

Now,

And

Let be the arithmetic mean of and

. Hence proved.

Question 9: Insert five numbers between and such that the resulting sequence is an A.P..

Answer:

Let be the 5 A.M.s between and .

Therefore are in A.P.

We know

Therefore

Therefore the 5 A.M.s between and are