Find the sum of the series
 to infinity

Sum to n terms
 

Sum to n terms, the series 1, 6, 15, 28,  .  .  .  .  .

Solve the equation
 
 

Sum the series
 

Sum the series
6 + 9 + 14 + 23 + 40 + .  .  .  .  .  to n terms

Sum the series
 to n terms

Sum of the series
 to infinity

Show that if a, b, c, d be four positive unequal quantities and  then .

Sum of the series

Sum of the series
 to infinity

If ,  are respectively arithmetic means of squares and cubes of all the numbers less than n and prime to it, prove that
 

If two roots of the equation  be equal and of opposite sign, show that pq = r.

Sum the following series to n terms
 

Solve the equation  having one root as

Form an equation whose roots shall be the product of every two roots of

Two numbers a and b are given, two other numbers and  are formed by the relation ,  Two more ,  are formed from ,  in the same manner and so on. Find  in terms of a and b and prove that when n is infinite  

If a, b, c are real positive quantities, show that
 

If  , prove that a, b, c are in harmonic progression, unless b = a + c .

If  are any four consecutive coefficients of extended

binomial, prove that

A purchaser is to take a plot of land facing a street, the plot is to be rectangular and three times the frontage added to twice its depth is to be 26 meters. What is the greatest number of square meters he may take?

Show that

Having given  , expand x in form
   and show that