2.11 Special Series

There are some series whose sum can be expressed by explicit formulae. Such series are called special series . Numbers and Sequences Here we study some common special series like (i) Sum of first ‘ n ’ natural numbers (ii) Sum of first ‘ n ’ odd natural numbers. (iii) Sum of squares of first ‘ n ’ natural numbers.

(iv) Sum of cubes of first ‘ n ’ natural numbers. We can derive the formula for sum of any powers of first n natural numbers using the expression ( k k . That is to find k k k k ... we can use the expression k k .

. . Sum of first n natural numbers To find  n , let us consider the identity ( Where x = , , ,.....n – , n x = , x = , x = ,    − , n , ( Adding all these equations and cancelling the terms on the Left Hand side, we get, n + =    n = n n  n = n n . .

Sum of first n odd natural numbers  It is an A.P. with a = , d = and l S n =   n a l   S n = = n . . Sum of squares of first n natural numbers To find  n , let us consider the identity ( Where x = , , ,.....n – , n x = , x = , x = ,    − , n , ( Adding all these equations and cancelling the terms on the Left Hand side, we get, n +    n n  n n ...

+ n n n  n n . . Sum of cubes of first n natural numbers To find  n , let us consider the identity Where x = , , ,.....n – , n x = , x = , x = ,    − , n , ( Adding all these equations and cancelling the terms on the Left Hand side, we get, ( n + ) – =     n n n n   n n n n  n =  n n Ideal Friendship Consider the numbers and . Sum of the divisors of (excluding ) = + + + + + + + + + + = Sum of the divisors of (excluding ) = + + + + = .

Thus, sum of divisors of one number excluding itself is the other. Such pair of numbers is called Amicable Numbers or Friendly Numbers. and are least pair of Amicable Numbers. They were discovered by Pythagoras.

We now know more than million amicable pair of Numbers.