. 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