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.

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

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)