TeamGMATIFY
During the month of April, the average temperatures on day n in two cities A and B are denoted by TA(n) and TB(n) respectively. The average temperature in city A on any day is the average of the average temperatures of the previous two days in city A. The average temperature in city B on any day is the average of the average temperatures of the previous two days in city B. If TA(1) = 30, TA(2) = 35, TB(1) = 35 and TB(2) = 30, which of the following statements must be true?
I. TA(4) - TB(4) > TA(29) - TB(29)
II. TA(4) + TB(4) = TA(29) + TB(29)
III. |TA(n+1) - TA(n)| = |TB(n+1) - TB(n)| for n>3
A. II only
B. III only
C. I and II only
D. II and III only
E. I, II and III
Look at the two sequences. The first two terms are the same - they are just interchanged.
TA - 30, 35, 32.5, ...
TB - 35, 30, 32.5,...
Why will the fourth term be different? Because it is the avg of 35 and 32.5 in case of A and that of 30 and 32.5 in case of B. Avg is the middle value. Let's put these on the number line.
Attachment:
Screenshot 2022-08-25 at 1.58.02 PM.png [ 19.68 KiB | Viewed 16494 times ]
Each smaller line is the avg of previous two lines. Avg of 30 and 35 is 32.5, the green line common to both A and B. Next day avg for B is the left black line while that for A is the right black line. Next day avg for B is the left yellow line while for A is the right yellow line etc.
The avgs for each day for A and B are equidistant from 32.5.
I. TA(4) - TB(4) > TA(29) - TB(29)
T(4) avgs are given by the black lines. Subsequent avgs are all closer to each other (corresponding) than the black lines i.e. the red lines, the yellow lines etc have less distance between them that the black lines.
Hence TA(4) - TB(4) is greater than TA(n) - TB(n) for all subsequent values of n.
Hence TA(4) - TB(4) is greater than TA(29) - TB(29). Must be true.
II. TA(4) + TB(4) = TA(29) + TB(29)
Each set of avgs is equidistant from 32.5. So the sum of each set of avgs will be 65.
TA(4) + TB(4) = TA(29) + TB(29) = 65
Must be true
III. |TA(n+1) - TA(n)| = |TB(n+1) - TB(n)| for n>3
The distance between subsequent avgs on each side will be the same for all n > 3. The distance between black and yellow lines on the left is the same as the distance between black and yellow lines on the right. The distance between yellow and purple lines on the left is the same as the distance between yellow and purple lines on the right and so on...
Hence this must be true too.
Answer (E)