Consider, sum of the series 
f(i) f(j)
In the given summation, i and j are not independent.
In the sum of series 
= 
i and j are independent. In this summation, three types of terms occur, those when i < j, i > j and i = j.
Also, sum of terms when i < j is equal to the sum of the terms when i > j if f(i) and f(j) are symmetrical. So, in that case
= 
+ 
+ 
= 2 
+ 
⇒ 
= 
When f(i) and f(j) are not symmetrical, we find the sum by listing all the terms.
(i) 
is equal to
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) 
= 
= 
= 
= 
(ii) 
= 
= 
= 3 n
(iii) . 
= 
= (n + 1)2 n – 2 n = n2 n
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