Prove that the sum of two increasing functions on a certain open interval is a function monotonically increasing on this interval.

The sum of two increasing functions on a certain open interval is a function monotonically increasing on this interval. Will the difference of increasing functions be a monotonic function?

Prove that if the graph of the function y = f (x), defined throughout the number scale, is symmetric about two vertical axes x = a and x = b (a < b), then this function is a periodic one.

The third term of Geometric Progression is 4. The product of the five terms is

a)

b)

c)

d)

The rational number which equals the numbers with recurring decimals is

a)

b)

c)

d)

If a, b, c are in Geometric Progression then the equations  
have a common root if  are in

a) Arithmetic Progression

b) Geometric Progression

c) Harmonic Progression

d) None of these

If a, b, c, d and p are distinct real numbers such that
 
then a, b, c, d

a) Are in Arithmetic Progression

b) Are in Geometric Progression

c) Are in Harmonic Progression

d) Satisfy ab = cd

e) Satisfy none of these

Sum of the first n terms of the series  is

a) 2ⁿ – n – 1

b) 1 – 2^{− n}

c) n + 2^{− n} – 1

d) 2ⁿ + 1

If  are in Arithmetic Progression
then

a) a, b, c are in Arithmetic Progression

b)  are in  Arithmetic Progression

c) a, b, c are in Geometric Progression

d) a, b, c are in Harmonic Progression

Let T_r be the r^th term of an Arithmetic Progression for  If for some positive integers m, n we have  and  then T_mn equals

a)

b)

c)

d)

If  are in Geometric Progression then
 are in

a) Arithmetic Progression

b) Geometric Progression

c) Harmonic Progression

d) None of these

Let  be in Arithmetic Progression and
 be in Harmonic Progression. If  and
 then  is

a) 2

b) 3

c) 5

d) 6

The harmonic means of the roots of the equation
 is

a) 2

b) 4

c) 6

d) 8

Consider an infinite geometric series with first term a and common ratio r. If its sum is four and the second term is  then

a)

b)

c)

d)

Let the positive numbers a, b, c, d be in Arithmetic Progression. Then
abc, abd, acd, bcd are

a) Not in Arithmetic Progression/Geometric Progression/Harmonic Progression

b) In Arithmetic Progression

c) In Geometric Progression

d) In Harmonic Progression

The sum of the first 2n terms of the Arithmetic Progression 2, 5, 8, . . . .  is equal to the sum of the first n terms of the Arithmetic Progression 57, 59, 61, . . . . then n equals

a) 100

b) 12

c) 11

d) 13

If  are positive real numbers whose product is a fixed number c then the minimum value of
  is

a)

b)

c)

d)

Suppose a, b, c are in Arithmetic Progression and  are in Geometric Progression. If  then the value of a is

a)

b)

c)

d)

An infinite Geometric Progression has first term x and sum 5 then

a)

b)

c)

d)

If α, β are roots of  and  are in Geometric Progression and  then

a)

b)

c)

d)

Multiple Choice

For  if
 , then

a)  

b)  

c)  

d)  

If x, y, z are in Harmonic Progression then show that
 

If the m^th, n^th and p^th term of an Arithmetic Progression and a Geometric Progression are equal and are x, y, z then prove that