The angles of a triangle are in Arithmetic Progression and let . Find the angle A.

If  are in Arithmetic Progression where  for all i, show that
 

If n is a natural number such that  and  are distinct primes, then show that
 

The sum of the squares of three distinct real numbers which are in Geometric Progression is . If their sum is , show that
 

If  are in Arithmetic Progression, determine the value of x.

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of four consecutive terms of it, prove that the resulting sum is square of an integer.

If a, b, c are in Arithmetic Progression and are in Harmonic Progression then prove that either or a, b and  are in Geometric Progression.

If a, b, c are positive real numbers then prove that
 

Let the Harmonic Mean and Geometric Mean of two positive numbers be in the ratio of 4:5. Then the two numbers are in the ratio .  .  .  .  .

Let p, q be the roots of the equation , and r and s are roots of the equation . If  are in arithmetic progression then A = .  .  .  .  . , B = .  .  .  .  .

Let x be the Arithmetic Mean and y, z be two Geometric Means between any two positive numbers then

Given positive integers r > 1, n > 2 and the coefficients of (3r)^th term and (r + 2)^th terms in the binomial expansion of (1 + x)²ⁿ are equal then

a) n = 2r

b) n = 2r + 1

c) n = 3r

d) none of these

Coefficient of x⁴ in  is

a)

b)

c)

d) None of these

The product of n positive real numbers is unity. Then their sum is

a) A positive integer

b) Divisible by n

c) Equal to

d) Never less than n

If in the expansion of (1 + x)^m (1 – x)ⁿ, the coefficients of x and x² are 3 and –6 respectively. then m is

a) 6

b) 9

c) 12

d) 24

In the binomial expansion of  the sum of the 5^th term and 6^th term is zero, then  equals

a)

b)

c)

d)

Coefficient of t²⁴ in (1 + t²)¹² (1 + t¹²) (1 + t²⁴) is

a)

b)

c)

d)

Prove that  is divisible by 25 for any natural number n.

Use mathematical induction to prove that  is divisible by 24 for all n > 0.

Find the sum of the series
 

Prove that C₀ – 2²C₁ + 3²C₂ − .  .  .  + (−)ⁿ  (n + 1)² C_n = 0 for n > 2 where

Find the natural number a for which
 
where the function f satisfies the relation f (x + y) = f (x).f(y)for all natural numbers x and y and further f (1) = 2

Using mathematical induction prove that