2015/058) 100 positive integers are written in a row. The average of the first and second numbers is 1. The average of the second and third numbers is 2. The average of the third and fourth numbers is 3. This pattern continues, and the average of the 99th and 100th number is 99. What is the 100th number?

Average of $1^{st}$ and $2^{nd}$ number is 1 so sum is 2 so both positive integers have to be 1

so $a_1= 1$
$a_2 = 1$

now average of $2^{nd}$ and $3^{rd}$ is 2 so sum = 4 so $a_3= 3$

average of $3^{rd}$ and $4^{th}$ is 3 so sum = 6 so $a_4= 3$

average of $4^{th}$ and $5^{th}$ is 4 so sum = 6 so $a_5= 5$

so we have $n^{th}$ term is n when n is odd and n-1 when n is even

this can be proved as below

for n odd 2 numbers are n and n and average = n
for n even the number is n-1 and next number is n+1 and average is n

so $100^{th}$ number = 99

refer to https://in.answers.yahoo.com/question/index?qid=20121112041716AA4V52E