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 ?

Options

A. `0`
B. `9054`
C. `12072`
D. `6036`

Answer

D

Explanation

`(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`