Question

Suppose $a_1,a_2,a_3,\dots ,a_{2012}$ are integers arranged on a circle. Each number is equal to the average of its two adjacent numbers. If the sum of all even indexed numbers is $3018$, what is the sum of all numbers?

• A. $0$
• B. $1509$
• C. $3018$
• D. $6036$

Answer 1

Answer: (D)

$6036$

Each number is average of two adjacent numbers.

$\therefore a_2=\frac{a_1+a_3}{2}$

$a_4=\frac{a_3+a_5}{2}$

$....$

$a_{2012}=\frac{a_{2011}+a_1}{2}$

Adding these equations, we have

$a_2+a_4+...+a_{2012}=\frac{2(a_1+a_3+a_5+...+a_{2011})}{2}$

Thus, the sum of even indexed and odd indexed numbers is the same.

So, the total sum of all the numbers in the group would be $2\times 3018=6036$