If ,  are respectively arithmetic means of squares and cubes of all the numbers less than n and prime to it, prove that
 

Find the n^th term and the sum of n terms of the recurring series  to n terms.

Sum the following series to n terms
 

If  , prove that a, b, c are in harmonic progression, unless b = a + c .

Having given  , expand x in form
   and show that
 

Prove if a, b, c are unequal and in harmonic progression, then
 

Prove if a, b, c are in harmonic progression, then
 

If the sum of m terms of an arithmetic progression is equal to sum of either the next n terms or the next p terms, prove that
 

A geometric and harmonic progression have the same p^th, q^th, r^th terms a, b, c respectively. Show that

The area of triangle with vertices (a, b),  and  where  are in geometric progression with common ratio r, and b are in geometric progression with common ratio s is given by

a)

b)

c)

d) None of the above

Let   , , 2, 3 then  is not greater than

a)

b) 1

c) 2

d) 4

The series  is equal to

a)

b)

c)

d)

The sum of n terms of three arithmetic progressions whose first term is 1 and common differences 1, 2, 3 are  respectively. The true relation is

a)

b)

c)

d)

The arithmetic mean, harmonic mean and geometric mean between two numbers are , 15 and 12 but not necessarily in this order. Then harmonic mean, geometric mean and arithmetic mean respectively are

a) 15, 12,

b) , 12, 15

c) 12, 15,

d) , 15, 12

Statement 1 : If 3, 6, 12 are in geometric progression then 9, 12, 18 are in harmonic progression
Statement 2 : If middle term is added in three consecutive terms of a geometric progression, resultant will be in harmonic progression

a) Statement 1 is false. Statement 2 is true

b) Statement 1 is true. Statement 2 is true. Statement 2 is correct explanation for statement 1

c) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation for statement 1

d) Statement 1 is true. Statement 2 is false

If  terms of an arithmetic progression are in geometric progression, and m, n, r are in harmonic progression then the value of the ratio of the common difference to the first term of the arithmetic progression is

a)

b)

c)

d)

The sum of three terms of a strictly increasing geometric progression is αS and the sum of squares of these terms is .  lies in

a)

b)

c)

d) None of these

The sum of three terms of a strictly increasing geometric progression is αS and the sum of the square of these terms is . If  then the greatest value of the middle term is

a) 5

b)

c) 10

d)

The sum of n terms of the following series
  will be

a)

b)

c)

d) None of these

If  and  are in arithmetic progression then the value of x is

a) 0

b) −1

c) 3

d) None of these

If the sum of the series  is a finite number then

a)

b)

c)

d)

If   then  equals

a)

b)

c)

d) None of these

Prove that if for a linear function f (x) = ax + b, the value of the argument x = (n = 1, 2, 3.  .  .) form an arithmetic progression then the corresponding value of  the function   = f () (n = 1, 2, 3, ...) also form an arithmetic progression.