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)

Multiple choices

If the first and  term of an Arithmetic Progression, a Geometric Progression and a Harmonic Progression are equal and their n^th term are a, b, c respectively then

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.

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