Ifa1+a2+...+an=0a12+a22 +..+an2 =1then what would bea1a2+a2a3+a3a4+....+ana1= ?
Raman , 7 Years ago
Grade 12th pass
Magical Mathematics[Interesting Approach]> If a1+a2+...+an=0 a1 2 +a2 2 +..+an 2 =1 ...
1 Answersdeepak
Last Activity: 7 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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