Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of the 7th and (m – l)th numbers is 5 : 9. Find the value of m.

For what value of n, is the A.M. between a and b?

Prove that the sum of n arithmetic means between two numbers is n times the single A.M. between them.

The sum of two numbers is . An even number of arithmetic means are being inserted between them and their sum exceeds their number by 1. Find the number of means inserted.

Insert A.M.’s between 7 and 71 in such a way that the 5th A.M. is 27. Find the number of A.M.’s.

The sum of three numbers in A.P. is 21 and the product of first and the third numbers exceeds the second number by 6, find the three numbers.

Find four numbers in A.P. whose sum is 32 and the product of extremes is to the product of means is 7 : 15.

If the lengths of the sides of a right angled triangle are in A.P., then show that their ratio is 3 : 4 : 5.

The digits of a positive integer having three digits are in A.P. and their sum is 15.

The number obtained by reversing the digits is 594 less than the original number.

Find the number.

50.

If and are in A.P., Prove that are also in A.P.