KVPY SX SB Exam 2012 Question Paper

Question : 19 of 120

Marks: +1, -0

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

Validate

Solution:

a_1, a_2, a_3, …….,a_2012 = 3018

…………………(1)

{a_1+a_3}/2=a_2

2a_2+2a_4+2a_6+...+2a_2012=6036

(a_1+a_3)+(a_3+a_5)

+(a_5+a_7)+....

+(a_2011+a_1)=6036

2(a_1+a_3+a_5+...+a_2011)=6036

a_1+a_3+a_5+...+a_2011=3018

...........................................(2)
Add (1) and (2)
Sum of all number = 3018 + 3018 = 6036

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