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.

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

Find the sum of the series
 

Find the sum to n terms

a)

b)

c)

d)

Sum to n terms 1.5 + 2.6 + 3.7 + .  .  .  .

a)

b)

c)

d)

Sum to n terms

a)  

b)  

c)  

d)  

Sum to n terms  

a)  

b)  

c)  

d)  

Sum to n terms 1.2.4 + 2.3.5 + 3.4.6 + .  .  .  .

a)  

b)  

c)  

d)  

Sum to n terms 1.2.3 + 4.5.6 + 7.8.9 + .  .  .  .

a)  

b)  

c)  

d)  

Sum to n terms 2.1 + 5.3 + 8.5 + .  .  .  .

a)  

b)  

c)  

d)  

Find the value of

a) 2926

b) 1826

c) 2026

d) 2126

Sum to n terms 1.n+2.(n – 1) + 3.(n – 2) + .  .  .  .

a)  

b)  

c)  

d)  

Find the sum of the products, taken two together of the first n natural numbers.

a)  (3n + 2)

b)  (3n + 2)

c)  

d)  

Sum to n terms

a)  

b)  

c)  

d)  

Sum to n terms

a)  

b)  

c)  

d)  

Sum to n terms

a)  

b)  

c)  

d)  

Sum to n terms

a)

b)

c)

d)

Sum up to n terms 1.1! + 2.2! + 3.3! + 4.4! + .  .  .  .

a) n! – 1

b) (n + 1)! – 1

c) (n +2)! – 1

d) (n – 1)! + 1

Sum up to n terms  

a)

b)

c)

d)

Sum to n terms

a)

b)

c)

d)

Sum up to n terms and up to infinity
 

a)

b)

c)

d)