If x, y, z are in harmonic progression show that
 

If x, y, z are such that their sum is constant and if
( z + x – 2y ) ( x + y – 2z ) varies as yz,
prove that 2 ( y + z ) – x varies as yz.

If , show that
 

Solve the equation
 

Given ln2 = 0.30103 and ln3 = 0.47712. Solve the equation
 

Show that the coefficient of  in the expansion of
    is    
according as n is of the form 3m, 3m + 1, 3m + 2.

If  is greater than 5x – 1 and less than 7x – 3, find the integral value of x.

Find the value of the infinite series
 

Show that the coefficient of  in the expansion of  in ascending powers of x is

Find the generating function, the sum to n terms and the n^th term of the recurring series

If a, b, c are all real quantities and  is divisible by
(x – a) and also by (x – b), prove that either
a = b = c or a = – 2b = – 2c.

Show that the sum of squares of three consecutive odd numbers increased by 1 is divisible by 12 and not by 24.

Show that  is greatest or least value of  according as a is negative or positive

If  and  is not greater than 1, show that
 

If
Prove that
 

If a > b > 0 and n is a positive integer prove that
 

Find the sum of the first n terms of the series whose r^th term is
 

Solve

If s is the sum of n positive unequal quantities then show that
 

If α, β are roots of   
and γ, δ are roots of  show that
 

In England one person out of 46 is said to die every year and one out of 33 is born. If there were no emigration, in how many years would the population double itself at this rate.

Find the sum to infinity of the series whose n^th term is
 

Find the sum of the series
6, 24, 60, 120, 210, 336, . . . . . . to n terms.