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Sum to n terms of Special Series



Let us find the values (formula of (1), (2) and (3)


This is an A.P with .



Adding vertically both sides,


(3) Let

Let us consider the pattern:



Note:
All the formulae for can be proved by mathematical induction.

Example 1:
Find the sum to terms of the series whose term is

Solution:




Example 2:
Find the sum of 20 terms of the series

Solution:
term of the series
=(term of the A.P 1,2,3,..) (term of the series )


Example 3:
Find the sum to terms of the series 3+7+13+21+31+............

Solution:


Subtracting



Example 4:
Find the sum to n terms of the series :

Solution:


adding vertically,





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