Suppose `a_(1)`,`a_(2)`,`a_(3)`,….,`a_(2012)` are integers arranged on a cicle. Each number is equal to the average of its two adjacent numbers. If the sum of all even idexed numbers is `3018`, what is the sum of all numbers ?
A
`0`
B
`9054`
C
`12072`
D
`6036`
Text Solution
Verified by Experts
The correct Answer is:
D
`(d)` `a_(2)=(a_(1)+a_(2))/(2)` `a_(3)=(a_(2)+a_(4))/(2)` `a_(1)=(a_(2)+a_(2012))/(2)` `a_(2012)=(a_(2001)+a_(1))/(2)` Now `a_(2)+a_(4)+…+a_(2012)=3018`…….`(i)` `2a_(2)+2a_(4)+..+2a_(2012)=6036` `:.a_(1)+a_(2)+a_(3)+a_(5)+...+a_(2011)+a_(1)=6036` `:.2(a_(1)+a_(3)+...+a_(2011))=6036` `:.a_(1)+a_(3)+...+a_(2011)=3018`.........`(ii)` By adding `(i)` and `(ii)` we get `a_(1)+a_(2)+a_(3)+...+a_(2012)=6036`
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