Q12/117 )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 if 1st and 2nd number is 1 so sum is 2 so both positive integers have to be 1

so a1= 1
a2 = 1

now average of 2nd and 3rd is 2 so sum = 4 so a3= 3

average of 3rd and 4th is 3 so sum = 6 so a4= 3

average of 4th and 5th is 4 so sum = 6 so a5= 5

so we have nth 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 100th number = 99