Class 11

Math

Algebra

Sequences and Series

Suppose $a_{1},a_{2},a_{3}........a_{2012}$ 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?

- $0$
- $1509$
- $3018$
- $6036$

$S_{n}=a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+......+a_{2012}$

Number of even terms $=22012 =1006$

Number of odd terms $=1006$

Number of odd terms $=1006$

$∣a_{1}∣=∣a_{2}∣=∣a_{3}∣=∣a_{4}∣=......=∣a_{2012}∣$

$3018=a_{2}+a_{4}+a_{6}+a_{8}+......+a_{2012}3018=1006∣a∣3=∣a∣$

Total $Sum=a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+......+a_{2012}S_{n}=(2012)∣a∣S_{n}=3×2012S_{n}=6036$