The sum of first n terms of the series  +  +  +  + .  .  .  . is equal to

a) 2ⁿ – n + 1

b) 1 – 2^-n

c) n + 2^-n – 1

d) 2ⁿ – 1

If the (m + 1)th, (n + 1)th and the (r + 1)th term 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 value of . Given that a, x, y, z, b are in Arithmetic Progression while the value of . If a, x, y, z, b are in Harmonic Progression then find a and b.

The interior angles of a polygon are in Arithmetic Progression. The smallest angle is 120° and the common difference is 5. Find the number of sides of the polygon.

Does there exist a Geometric Progression containing 27, 8 and 12 as three of its terms? If it exists, how many such progressions are possible?

If S₁, S₂, .  .  .  .,S_n are the sums of infinite geometric series whose first terms are 1, 2, 3,   .  .  ., n and whose common ratios are  respectively, then find the value of

The real numbers x₁, x₂, x₃ satisfying the equation x³ – x² + βx + γ = 0 are in Arithmetic Progression. Find the interval in which β and γ lie.

Let a₁, a₂, … a_n be positive real numbers in Geometric Progression. For each n let A_n, G_n, H_n be respectively the arithmetic mean, geometric mean and harmonic mean of a₁, a₂, .  .  .  ., a_n. Find the expressions for the Geometric mean of G₁, G₂, .  .  .  .G_n in terms of A₁, A₂, .  .  .  .,A_n; H₁, H₂, .  .  .  .H_n

Let a, b are real positive numbers. If a, A₁, A₂, b are in Arithmetic Progression, a, G₁, G₂, b are in Geometric Progression and a, H₁, H₂, b are in Harmonic Progression show that
 

If total number of runs scored in n matches is
 where n > 1 and the runs scored in the k^th match are given by k.2^{n + 1 – k}  where 1 ≤ k ≤ n. Find n.

If three complex numbers are in Arithmetic Progression, then they lie on a circle in a complex plane.

a) True

b) False

The sum of integers from 1 to 100 that are divisible by 2 or 5 is

In a triangle ABC if cotA, cotB, cotC are in Arithmetic Progression then a, b, c are in .  .  .  .  . Progression.

For any odd integer n ≥ 1,
n³ – (n – 1)³ + .  .  . + (−)^{n – 1} 1³ = .  .  .

If  be two sequences given by  for all n  then  is equal to

a)

b)

c)

d)

If  are in arithmetic progression whose common difference is α then the value of  is equal to

a)

b)

c)

d)

If  be two arithmetic means and  be two geometric means between a and b then  is equal to

a)

b)

c)

d)

A person is to count 4500 currency notes. Let denotes the number of notes he counts in the  minutes. If  and   are in arithmetic progression with common difference –2 , then the time taken by him to count all notes is

a) 24 minutes

b) 34 minutes

c) 125 minutes

d) 135 minutes