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If a1+a2+...+an=0 a1 2 +a2 2 +..+an 2 =1 then what would be a1a2+a2a3+a3a4+....+ana1= ?

If
a1+a2+...+an=0
a12  +a22 +..+an2 =1
then what would be
a1a2+a2a3+a3a4+....+ana1= ?

Grade:12th pass

1 Answers

deepak
84 Points
5 years ago
consider,
(a1+a2+a3...+an)2= a12 +..an2 +2(sum of all the posible pairs like aiaj where i,j belongs (1,2,3...n)  and i=j)=0
therefore
(sum of all the posible pairs like aiaj where i,j belongs (1,2,3...n)  and i=j)= −½
 
now consider the given series,
here the second term such as in a1a2 can be substituted as like a2= −(a1+a3...an) from the eqn 1 given in the question
 
hence the overall series will become like,
= −{(a12 + a22+...an2)+(n−1)(sum of all the posible pairs like aiaj where i,j belongs (1,2,3...n)  and i=j)}
=−{1+(1−n)/2}
=(n−1)/2  − 1
 
 
hope this is useful

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